diff --git a/build/a.html b/build/a.html index 3ae8e8f..8d599ee 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    diff --git a/build/a_l.html b/build/a_l.html index 1465dae..4cc6c61 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1727,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c.html b/build/c.html index 2151581..bf3331e 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m.html b/build/c_m.html index f129529..2344296 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1650,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_cs.html b/build/c_m_cs.html index d231674..22e151a 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index 5742373..b699323 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1668,14 +1643,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Gradient
    -
    +

    -
    +
    • Gr4(1.79)
    @@ -1696,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Divergence
    -
    +

    -
    +
    • Gr4(2.21)
    @@ -1724,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
    Curl
    -
    +

    -
    +
    • Gr4(2.21)
    @@ -1778,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index 6521065..64e883a 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1639,7 +1614,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index c06e092..1bfac1b 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1752,7 +1727,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc.html b/build/c_m_dc.html index e5a75c3..2f3a04c 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1649,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index 0fb3e77..dda8fdd 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1651,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index b9024f9..0df3b18 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1624,9 +1599,9 @@ Table of contents
    -
    -
    Divergence of gradient
    -
    +
    +
    Divergence of gradient
    +

    \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the Laplacian of the scalar field \(T\). 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.

    -
    -
    Curl of a gradient
    -
    +
    +
    Curl of a gradient
    +

    This always vanishes.

    -
    -
    Gradient of the divergence
    -
    +
    +
    Gradient of the divergence
    +

    \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.

    -
    -
    Divergence of a curl
    -
    +
    +
    Divergence of a curl
    +

    This always vanishes.

    -
    -
    Curl of curl
    -
    +
    +
    Curl of curl
    +

    \[ {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v} @@ -1689,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 6a4b586..31ce4e4 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1650,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index bc3c231..e18760d 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1647,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 8028591..131417f 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1671,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 8ceea35..449a828 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dd.html b/build/c_m_dd.html index 29dadec..d6dff68 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index c51e9e1..a258f42 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1658,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 1e4b601..821670e 100644 --- a/build/c_m_dd_3d.html +++ b/build/c_m_dd_3d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox: More generally,

    -
    +

    -
    +
    • Gr (1.100)
    @@ -1671,9 +1646,24 @@ More generally, Since

    +
    +

    + + + + +

    +
    +
      +
    • Gr (1.101)
      • +
    + +
    + +
    \begin{equation} {\boldsymbol \nabla}_1 \left(\frac{1}{r_{12}}\right) = -\frac{\hat{\bf r}_{12}}{r_{12}^2} -\label{Gr(1.101)} +\tag{div1or}\label{div1or} \end{equation}

    @@ -1701,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index 8de4d8c..65bf3cf 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1664,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic.html b/build/c_m_ic.html index df1b55c..bad6498 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1649,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index bc8397a..430bac4 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1647,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index bf77198..617c800 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1656,7 +1631,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index 151b85b..43463c4 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1650,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index 36df98f..b85e2a1 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1657,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 5a02560..da8ccb1 100644 --- a/build/c_m_ic_lsv.html +++ b/build/c_m_ic_lsv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1624,9 +1599,9 @@ Table of contents
    -
    -
    Line Integrals
    -
    +
    +
    Line Integrals
    +

    \[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} @@ -1655,9 +1630,9 @@ Integral over a closed loop:

    -
    -
    Surface Integrals
    -
    +
    +
    Surface Integrals
    +

    \[ \int_{\cal S} {\bf v} \cdot d{\bf a} @@ -1677,9 +1652,9 @@ Over a closed surface:

    -
    -
    Volume Integrals
    -
    +
    +
    Volume Integrals
    +

    \[ \int_{\cal V} T d\tau @@ -1718,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index 6a5c710..757ac4e 100644 --- a/build/c_m_ic_stokes.html +++ b/build/c_m_ic_stokes.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1622,14 +1597,14 @@ Table of contents c.m.ic.stokes
    -
    +

    -
    +
    • Gr (1.57)
    @@ -1672,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1659,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1659,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 79a59d7..88e40db 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1662,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1648,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1677,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1672,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index e69afd3..1ffdad7 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1706,7 +1681,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1672,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index 892d4f9..40b8bf8 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1719,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_vf.html b/build/c_m_vf.html index f795bf6..af3c8fb 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index 209247a..a510b43 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1664,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index 380d3d9..7cbcfa2 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1691,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d.html b/build/d.html index 25189eb..c535834 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents d
    -
    - +
    + Objectives @@ -1686,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1666,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index 8d4d826..67fab85 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1661,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index 27e4653..0aeabd4 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1664,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_emf.html b/build/d_emf.html index 33849c1..c13c3e9 100644 --- a/build/d_emf.html +++ b/build/d_emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1661,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_ems.html b/build/d_ems.html index 9222c64..4b9babe 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1683,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index f139b6c..6e6327c 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1666,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 0a7169b..5b74bf8 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1663,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_emsm.html b/build/d_emsm.html index e4f6589..e7a1a6b 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
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  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1666,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index 37a84ef..6d4822c 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1664,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/d_m.html b/build/d_m.html index bd56567..636c2a2 100644 --- a/build/d_m.html +++ b/build/d_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1674,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1668,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emd.html b/build/emd.html index 939d834..a904065 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emd
    -
    - +
    + Prerequisites
      @@ -1633,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1673,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 5263459..f9c00dc 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1647,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index e028556..d5a26c8 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1663,7 +1638,7 @@ Empirically: the changing magnetic field induces an electric current around the circuit. This current is really driven by an electric field having a component along the wire. The line integral of this field is called the

    -
    +

    Electromotive force (or electromotance), \[ @@ -1685,7 +1660,7 @@ to the rate of change of the magnetic flux, \] so we obtain

    -
    +

    Faraday's law (integral form N.B.: for a stationary loop) \[ @@ -1703,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

    -
    +

    Faraday's law (differential form) \[ @@ -1738,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index 198853f..9fc8f5f 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1682,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold \] We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with

    -
    +

    \[ 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 \hspace{2cm} \mbox{(7.31 and 7.34)} \end{align} -

    +

    \paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\). Find energy stored in section of length \(l\). @@ -1740,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 560cf1a..c8cb1e0 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1662,7 +1637,7 @@ M_{12} = M_{21} \]

    -
    +

    \paragraph{Example 7.10:} short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside @@ -1712,7 +1687,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).

    -
    +

    \paragraph{Example 7.11:} find self-inductance of toroidal coil with rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\)) @@ -1739,7 +1714,7 @@ Total flux: \(N\) times this, so self-inductance is Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.

    -
    +

    \paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). What is the current ? @@ -1776,7 +1751,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index 1146247..2b8fc4d 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ law in integral form: -
    +

    {\bf Example 7.7:} \({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? @@ -1662,7 +1637,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.

    -
    +

    {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim. Uniform magnetic field \({\bf B}_0\) in central region up to \(a < b\), @@ -1696,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with 'slow enough' phenomena.

    -
    +

    {\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find induced \({\bf E}\) field as a function of distance \(s\) from wire. @@ -1742,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Me.html b/build/emd_Me.html index 39a31a7..7ead01c 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1647,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index 3e5a2b9..778cf94 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,7 +1600,7 @@ Table of contents

    Full set of equations for the electromagnetic field:

    -
    +

    {\bf Maxwell's equations} {\it (in vacuum)}

    @@ -1641,7 +1616,7 @@ Full set of equations for the electromagnetic field:

    Complement:

    -
    +

    {\bf Force law} \[ @@ -1665,7 +1640,7 @@ take divergence of \((iv)\).

    Better way of writing: all fields on left, all sources on right,

    -
    +
    \begin{align} (i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, &(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\ @@ -1693,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index 84d2c53..e4246e1 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1632,7 +1607,7 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

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    +

    \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} @@ -1656,7 +1631,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

    Maxwell baptized this term the

    -
    +

    {\bf Displacement current} \[ @@ -1702,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index f76a4d4..d1fbbea 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1672,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index 029d379..a1e6f70 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1670,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce.html b/build/emd_ce.html index 9ebbad0..f96d99b 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emd.ce
    -
    - +
    + Prerequisites
      @@ -1632,8 +1607,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1669,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 50eb79f..2fe0b19 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,7 +1600,7 @@ Table of contents

    The angular momentum of EM fields is directly given by

    -
    +

    {\bf Angular momentum of EM fields} \[ @@ -1653,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index 55fc040..1cf6377 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

    -
    +

    {\bf Continuity equation} \[ @@ -1683,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index 825afe8..713e3ed 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1637,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM

    This is thus simply a conservation law for momentum, with

    -
    +

    {\bf Momentum density in the EM fields} \[ @@ -1649,7 +1624,7 @@ This is thus simply a conservation law for momentum, with

    In a region in which the mechanical momentum is not changing due to external influences, we then have the

    -
    +

    {\bf Continuity equation for EM momentum} \[ @@ -1676,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index 1fdd3be..5ee898d 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1679,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get

    This expression can be greatly simplified by introducing the

    -
    +

    {\bf Maxwell stress tensor} \[ @@ -1702,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction

    We then obtain

    -
    +

    {\bf EM force per unit volume} \[ @@ -1714,7 +1689,7 @@ We then obtain

    where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the

    -
    +

    {\bf Total force on charges in volume} \[ @@ -1741,7 +1716,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index 8c3dd2a..855f3ce 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
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  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1691,7 +1666,7 @@ so we get Substituting this in \ref{Gr(8.6)} and using the divergence theorem, we obtain

    -
    +

    {\bf Poynting's theorem} \[ @@ -1716,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.

    Energy per unit time, per unit area carried by EM fields:

    -
    +

    {\bf Poynting vector} \[ @@ -1729,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:

    We can thus express Poynting's theorem more compactly:

    -
    +

    {\bf Poynting's theorem} \[ @@ -1742,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:

    where we have defined the total

    -
    +

    {\bf Energy in electromagnetic fields} \[ @@ -1765,7 +1740,7 @@ Then, \] so we get the

    -
    +

    {\bf Poynting theorem (differential form)} \[ @@ -1782,7 +1757,7 @@ and has a similar for to the continuity equation -

    +

    \paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting. Assuming that the field is uniform, the electric field parallel to the wire is @@ -1825,7 +1800,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_emw.html b/build/emd_emw.html index 28f2f52..c4a744d 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emd.emw
    -
    - +
    + Prerequisites
      @@ -1633,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1673,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index e836bb4..0deb446 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1653,7 +1628,7 @@ so for a monochromatic EM plan wave, \] or more succinctly:

    -
    +

    {\bf Poynting vector of a monochromatic EM wave} \[ @@ -1669,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

    Similary, we get the

    -
    +

    {\bf Momentum density of a monochromatic EM wave} \[ @@ -1718,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index 787c470..2995cae 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1654,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. Generalizing to propagation in the direction of an arbitrary wavevector \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the

    -
    +

    {\bf E and B fields for a monochromatic EM plane wave} \[ @@ -1696,7 +1671,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index afb0e26..7b322b5 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1651,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \( Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), we get the

    -
    +

    {\bf Wave equations for electric and magnetic fields in vacuum} \[ @@ -1705,7 +1680,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    Created: 2022-02-09 Wed 07:31

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    diff --git a/build/emdm.html b/build/emdm.html index 532e030..95cd6ff 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 3cb351b..5765049 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 99df269..81aeab8 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1658,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

    -
    +

    {\bf Polarization current density} \[ @@ -1676,7 +1651,7 @@ the polarization current is the result of linear motion of charge when polarization changes). We can check consistency with the continuity equation associated to the conservation of bound charges:

    -

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emf.html b/build/emf.html index 6c19afd..701b5b0 100644 --- a/build/emf.html +++ b/build/emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emf
    -
    - +
    + Prerequisites
      @@ -1632,8 +1607,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1667,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emf_g.html b/build/emf_g.html index cf75550..ea51722 100644 --- a/build/emf_g.html +++ b/build/emf_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1664,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1663,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emf_g_Lg.html b/build/emf_g_Lg.html index 6fcfeb3..882287f 100644 --- a/build/emf_g_Lg.html +++ b/build/emf_g_Lg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1638,7 +1613,7 @@ while the equation for \(V\) becomes \] These can be written compactly upon introducing a new operator: the

    -
    +

    {\bf d'Alembertian operator} \[ @@ -1651,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the

    so we get the

    -
    +

    {\bf Inhomogeneous Maxwell equations (Lorenz gauge)} \[ @@ -1699,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/emf_svp.html b/build/emf_svp.html index 23ebda4..e2d7366 100644 --- a/build/emf_svp.html +++ b/build/emf_svp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1639,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.

    Easiest:

    -
    +

    \[ {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A} @@ -1655,7 +1630,7 @@ Putting this into Faraday's law gives \] so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get

    -
    +

    \[ {\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 \

    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

    -
    +

    \[ {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0} @@ -1684,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes \] which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),

    -
    +

    \[ \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.

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    diff --git a/build/ems.html b/build/ems.html index d9d9c8a..bcd5a3f 100644 --- a/build/ems.html +++ b/build/ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1647,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/ems_ca.html b/build/ems_ca.html index 27d0242..ff01d60 100644 --- a/build/ems_ca.html +++ b/build/ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
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  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents ems.ca
    -
    - +
    + Prerequisites
      @@ -1632,8 +1607,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1669,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html index 3159ba2..642bbf0 100644 --- a/build/ems_ca_fe.html +++ b/build/ems_ca_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1628,7 +1603,7 @@ A generic configuration of static charges coupled via the Coulomb interaction defines an electrostatic problem, whose solution is in principle obtained from calculating either the field according to E_vcd

    -
    +
    \begin{equation*} {\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} \end{equation*} @@ -1638,7 +1613,7 @@ from calculating either the field according to +
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1702,7 +1677,7 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f of the \(f_x + f_y + f_z = 0\) condition above.

    -
    +

    Earnshaw's theorem
    Since solutions to Laplace's equation have no local minimum, @@ -1803,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html index a9849ea..d03f561 100644 --- a/build/ems_ca_fe_g.html +++ b/build/ems_ca_fe_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1622,11 +1597,11 @@ Table of contents ems.ca.fe.g
    -
    +

    George Green

    -

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_fe_uP.html b/build/ems_ca_fe_uP.html index a50a6f0..56002cb 100644 --- a/build/ems_ca_fe_uP.html +++ b/build/ems_ca_fe_uP.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1652,7 +1627,7 @@ is unique. For Neumann, the solution is unique apart from an unimportant consta

    We can thus finally state the

    -
    +

    Uniqueness Theorem

    @@ -1712,7 +1687,7 @@ equipotentials, and capacitance relates the charge to the potential. }

    -
    +

    {\bf Comment/warning: {\color{blue} uniqueness theorem on uniqueness theorems}}
    Do not be misled by Griffiths: there is a {\it unique} uniqueness theorem for the @@ -1737,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    diff --git a/build/ems_ca_me.html b/build/ems_ca_me.html index 30eb138..40b2b4a 100644 --- a/build/ems_ca_me.html +++ b/build/ems_ca_me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1648,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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  • Field in terms of the potentialems.es.ep.fp -
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  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1667,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

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    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_me_Eq.html b/build/ems_ca_me_Eq.html index 0ebd5f9..8d0b3c6 100644 --- a/build/ems_ca_me_Eq.html +++ b/build/ems_ca_me_Eq.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_me_a.html b/build/ems_ca_me_a.html index 70fbafc..a321de7 100644 --- a/build/ems_ca_me_a.html +++ b/build/ems_ca_me_a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1710,7 +1685,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_me_h.html b/build/ems_ca_me_h.html index aa846ff..bf08f0e 100644 --- a/build/ems_ca_me_h.html +++ b/build/ems_ca_me_h.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1630,7 +1605,7 @@ V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^3} \in \] and can be rewritten as

    -
    +

    \[ V_{\mbox{\tiny quad}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} @@ -1641,7 +1616,7 @@ and can be rewritten as

    in terms of the {\bf quadrupole moment}

    -
    +

    \[ Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s). @@ -1659,7 +1634,7 @@ Moreover, it is traceless, \(\sum_a Q_{aa} = 0\). It therefore has \(5\) indepe

    Our expansion for the potential thus looks like

    -
    +

    \[ V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right) @@ -1678,7 +1653,7 @@ the leading nonvanishing multipole moment is independent of the chosen location the origin of the coordinate system (see Jackson Prob. 4.4).

    -
    +

    {\bf A consistent nomenclature for the multipole expansion?}\\\\ You all know the terms {\bf monopole}, {\bf dipole} and {\bf quadrupole}, @@ -1748,7 +1723,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_me_md.html b/build/ems_ca_me_md.html index d754632..3327ec9 100644 --- a/build/ems_ca_me_md.html +++ b/build/ems_ca_me_md.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1626,7 +1601,7 @@ Table of contents The series (\ref{Gr(3.95)}) is organized in increasing powers of inverse distance. The leading term is called the {\bf monopole} term, and is

    -
    +

    \[ V_{\mbox{\tiny mono}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r}, \hspace{1cm} @@ -1647,7 +1622,7 @@ V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{r^2} \int_ \] This can be written

    -
    +

    \[ V_{\mbox{\tiny di}} ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{\hat{\bf r} \cdot {\bf p}}{r^2}, @@ -1689,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_mi.html b/build/ems_ca_mi.html index 5d8dd76..e474af6 100644 --- a/build/ems_ca_mi.html +++ b/build/ems_ca_mi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1682,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_mi_fe.html b/build/ems_ca_mi_fe.html index dcc86ec..fbcdcfe 100644 --- a/build/ems_ca_mi_fe.html +++ b/build/ems_ca_mi_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1659,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_mi_isc.html b/build/ems_ca_mi_isc.html index 89a3845..fd5505f 100644 --- a/build/ems_ca_mi_isc.html +++ b/build/ems_ca_mi_isc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1654,7 +1629,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_mi_o.html b/build/ems_ca_mi_o.html index 21083d7..c891c96 100644 --- a/build/ems_ca_mi_o.html +++ b/build/ems_ca_mi_o.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1669,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_sv.html b/build/ems_ca_sv.html index 9b304c8..897453c 100644 --- a/build/ems_ca_sv.html +++ b/build/ems_ca_sv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1651,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_sv_car.html b/build/ems_ca_sv_car.html index c9affbd..8b4704e 100644 --- a/build/ems_ca_sv_car.html +++ b/build/ems_ca_sv_car.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1622,7 +1597,7 @@ Table of contents ems.ca.sv.car
    -
    +

    {\bf Example: separation of variables (Cartesian coordinates)}
    Two infinite grounded metal plates parallel to the \(xz\) plane, one at \(y = 0\) and the @@ -1723,7 +1698,7 @@ V(x,y) = \frac{4V_0}{\pi}\sum_{n=1, 3, 5, ...}^{\infty} \frac{1}{n} e^{-n\pi x/a

    -
    +

    {\bf Example: rectangular pipe}
    Two infinitely long grounded plates at \(y = 0,a\) are connected at \(x = \pm b\) @@ -1779,7 +1754,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_sv_cyl.html b/build/ems_ca_sv_cyl.html index c49d9fb..efce335 100644 --- a/build/ems_ca_sv_cyl.html +++ b/build/ems_ca_sv_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1640,7 +1615,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ca_sv_sph.html b/build/ems_ca_sv_sph.html index 33f647a..ea28cf0 100644 --- a/build/ems_ca_sv_sph.html +++ b/build/ems_ca_sv_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,7 +1600,7 @@ Table of contents

    In spherical coordinates, the Laplace equation takes the following form:

    -
    +

    @@ -1786,7 +1761,7 @@ Ex.: second solution for \(l = 0\) is

    General solution to problem with azimuthal symmetry:

    -
    +

    \[ V(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta) @@ -1798,7 +1773,7 @@ General solution to problem with azimuthal symmetry: -

    +

    Example: separation of variables (spherical coordinates)

    @@ -1890,7 +1865,7 @@ V(r, \theta) = \frac{k}{2} (1 - \frac{r}{R} \cos \theta).

    -
    +

    Example: surface charge density on sphere

    @@ -1935,7 +1910,7 @@ Since Legendre polynomials are orthogonal, B_l^> = A_l^< R^{2l + 1}. \label{Gr(3.81)} \] -Surface charge induces discontinuity in derivative of \(V\) from (\ref{Gr(2.36)}): +Surface charge induces discontinuity in derivative of \(V\) from dpdisc:

    @@ -2019,7 +1994,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es.html b/build/ems_es.html index 6898750..5c515b6 100644 --- a/build/ems_es.html +++ b/build/ems_es.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1629,8 +1604,8 @@ Table of contents
  • PM 1
    • -
    - +
    + Prerequisites
      @@ -1638,8 +1613,8 @@ Prerequisites
    -
    - +
    + Objectives
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1616,7 +1591,7 @@ Table of contents
    • - +

    Conductors @@ -1652,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_c_cap.html b/build/ems_es_c_cap.html index 675f97a..7cc96b3 100644 --- a/build/ems_es_c_cap.html +++ b/build/ems_es_c_cap.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents

  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,13 +1598,13 @@ Table of contents ems.es.c.cap

    -Two conductors, one at charge \(Q\) and the other at charge \(-Q\). Potential difference is -unambiguous: +Let us consider two conductors, one having charge \(Q\) and the other charge \(-Q\). +The potential difference is unambiguously given by

    \[ -V = V_+ - V_- = -\int_{(-)}^{(+)} {\bf E} \cdot d{\bf l} +V = \phi_+ - \phi_- = -\int_{(-)}^{(+)} {\bf E} \cdot d{\bf l} \]

    @@ -1644,10 +1619,14 @@ C \equiv \frac{Q}{V} \]

    -
    +

    -Example 2.11: -find the capacitance of two concentric spherical metallic shells with radii \(a\) and \(b\). +Concentric shells +

    + +

    +Consider two concentric spherical metallic shells with radii \(a\) and \(b\). +What is the capacitance of this system?

    @@ -1685,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_c_ic.html b/build/ems_es_c_ic.html index f70b4f9..090e48e 100644 --- a/build/ems_es_c_ic.html +++ b/build/ems_es_c_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1627,11 +1602,18 @@ Charge in hollow conductor. Induced surface charges on inside and outside surfaces.

    -
    +
    +

    +Field outside spherical conductor with irregular cavity +

    + +

    +Consider a spherical conductor centered at the origin, and having +an irregularly-shaped cavity containing a point charge \(q\) somewhere inside. +What is the field outside the sphere ? +

    +

    -Example 2.9: field outside a spherical conductor centered at origin, having -a cavity of a weird shape with a point charge \(q\) somewhere inside. What is the field -outside the sphere ? Answer: ${\bf E} = \frac{1}{4\pi\varepsilon_0} q \frac{\hat{\bf r}}{r^2}.$

    @@ -1663,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_c_p.html b/build/ems_es_c_p.html index 76e3b6d..c82c5c8 100644 --- a/build/ems_es_c_p.html +++ b/build/ems_es_c_p.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1622,7 +1597,7 @@ Table of contents ems.es.c.p
    -
    +

    Basic Properties of a Conductor

    @@ -1661,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_c_sc.html b/build/ems_es_c_sc.html index 5659f0c..9867d23 100644 --- a/build/ems_es_c_sc.html +++ b/build/ems_es_c_sc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,14 +1600,14 @@ Table of contents

    For a conductor, we can exploit the fact that electrical fields vanish on the inside to get a proper boundary condition for the potential. Namely, here, boundary condition -\ref{Gr(2.33)} yields +Edisc yields \[ {\bf E} = \frac{\sigma}{\varepsilon_0} \hat{\bf n} \label{Gr(2.48)} \] which in terms of potential reads \[ -\sigma = -\varepsilon_0 \frac{\partial V}{\partial n}. +\sigma = -\varepsilon_0 \frac{\partial \phi}{\partial n}. \label{Gr(2.49)} \] The force per unit area on the surface of an object is @@ -1650,7 +1625,8 @@ amounting to an outward electrostatic pressure. In terms of the field, P = \frac{\varepsilon_0}{2} E^2 \label{Gr(2.52)} \] -which can also be obtained from the principle of virtual work. +which can also be obtained from the principle of virtual work (considering the +change rate of energy with respect to volume).

    @@ -1671,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_e.html b/build/ems_es_e.html index 25d0c91..a0121fe 100644 --- a/build/ems_es_e.html +++ b/build/ems_es_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -429,30 +425,9 @@ Table of contents
    • -
  • - -
    - +
  • Electrostatic Energy from the Potentialems.es.e - -
    •  - -
  • @@ -1616,12 +1591,11 @@ Table of contents - +

    Electrostatic Energy from the Potential ems.es.e

    -
    • FLS II 8
      • @@ -1630,12 +1604,8 @@ Table of contents

    -We have already seen that the electrostatic potential is the potential energy of a unit charge brought from infinity. -

    - -

    -Since the force is proportional to the charge we're moving, -we can calculate the work per unit charge/ \(W_u\) (so \(W = q W_u\)), +We have already seen that the electrostatic potential is the potential energy of a unit charge brought from infinity, and that since the force is proportional to the charge we're moving, +we can simply calculate the work per unit charge \(W_u\) (so \(W = q W_u\)):

    @@ -1643,18 +1613,168 @@ we can calculate the work per unit charge/ \(W_u\) (so \(W = q W_u\)), W_u = -\int_{{\bf a}}^{{\bf b}} {\bf E} \cdot d{\bf l} \]

    + +

    +For a point charge distribution, this +can also be rewritten in terms of the potential, +

    +
    +

    + + + + +

    +
    +
      +
    • Gr (2.42)
      • +
    + +
    + +
    +

    +\[ +W = \frac{1}{2} \sum_{i=1}^m q_i \phi({\bf r}_i) +\tag{W_pcd}\label{W_pcd} +\] +

    + +

    +For a volume charge distribution, this becomes +

    + +
    +

    + + + + +

    +
    +
      +
    • Gr (2.43)
      • +
    + +
    + +
    +

    +\[ +W = \frac{1}{2} \int \rho({\bf r}) V({\bf r}) ~d\tau +\tag{W_vcd}\label{W_vcd} +\] +

    + +

    +In this, we can eliminate \(\rho\) and \(\phi\) in favour of \({\bf E}\): +

    + +

    +\[ +\rho = \varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E} +\longrightarrow +W = \frac{\varepsilon_0}{2} \int ({\boldsymbol \nabla} \cdot {\bf E}) ~\phi d\tau. +\] +

    + +

    +Integrating by parts and using \({\boldsymbol \nabla} \phi = -{\bf E}\), +

    + +

    +\[ +W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} \phi {\bf E} \cdot d{\bf a} \right) +\label{Gr(2.44)} +\] +

    + +

    +Integrating over all space, +

    +
    +

    + + + + +

    +
    +
      +
    • Gr (2.45)
      • +
    + +
    + +
    +
    +

    + +

    + +

    +\[ +W = \frac{\varepsilon_0}{2} \int E^2 d\tau +\tag{W_intEsq}\label{W_intEsq} +\] +

    + +
    + +
    +

    +Energy of spherical shell +

    + +

    +Consider a uniformly charged spherical shell of total charge \(q\) and radius \(R\). +We wish to compute its electrostatic energy. +

    + + +

    +first way: we use W_vcd adapted for surface charges, +\[ +W = \frac{1}{2} \int_S da \sigma \phi. +\] +

    + +

    +The potential at the surface of the sphere is constant, +\(\phi = \frac{1}{4\pi \varepsilon_0} \frac{q}{R}\), so +

    + +

    +\[ +W = \frac{1}{8\pi \varepsilon_0} \frac{q}{R} \int da \sigma = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}. +\] +

    + +

    +second way we use W_intEsq. Inside the sphere, \({\bf E} = 0\); outside, +

    + +

    +\[ +{\bf E} = \frac{1}{4\pi \varepsilon_0} q \frac{\hat{\bf r}}{r^2} \rightarrow E^2 = \frac{q^2}{(4\pi \varepsilon_0)^2 r^4}. +\] +

    + +

    +so +

    +\begin{align} +W &= \frac{\varepsilon_0}{2} \int_{outside~sphere} \frac{q^2}{(4\pi \varepsilon_0)^2 r^4} r^2 \sin \theta dr d\theta d\phi \nonumber \\ +&= \frac{1}{32 \pi^2 \varepsilon_0} q^2 4\pi \int_R^{\infty} \frac{dr}{r^2} = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}. +\end{align} + +
    +
    -
    In this section:
    - -
    Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_e_c.html b/build/ems_es_e_c.html deleted file mode 100644 index 5f41566..0000000 --- a/build/ems_es_e_c.html +++ /dev/null @@ -1,1655 +0,0 @@ - - - - - - -Pre-Quantum Electrodynamics - - - - - - - - - -
    -
    -

    -Pre-Quantum Electrodynamics -

    -
    - -
    -
    Comments on Electrostatic Energy - - -ems.es.e.c
    -
    -

    -(i) An 'inconsistency' ? No. Self-energy is simply excluded in \ref{Gr(2.42)}, but included -in \ref{eq:Energy_as_int_E2}. -(ii) Is the energy stored in the fields, or in the charges ? Meaningless question at this stage. -Radiation theory points to the preferable interpretation that the energy is stored in the field. -(iii) Superposition. Double the charge: quadruple the energy. -

    -
    -
    - - -
    -
    - -Creative Commons License - -Except where otherwise noted, all content is licensed under a -Creative Commons Attribution 4.0 International License. -
    -
    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    -

    -
    - - -
    - \ No newline at end of file diff --git a/build/ems_es_e_ccd.html b/build/ems_es_e_ccd.html deleted file mode 100644 index 10b4c39..0000000 --- a/build/ems_es_e_ccd.html +++ /dev/null @@ -1,1744 +0,0 @@ - - - - - - -Pre-Quantum Electrodynamics - - - - - - - - - -
    -
    -

    -Pre-Quantum Electrodynamics -

    -
    - -
    -
    The Energy of a Continuous Charge Distribution - - -ems.es.e.ccd
    -
    -

    -For this case, \ref{Gr(2.42)} becomes -

    - -

    -\[ -W = \frac{1}{2} \int \rho({\bf r}) V({\bf r}) ~d\tau -\label{Gr(2.43)} -\] -

    - -

    -Eliminate \(\rho\) and \(V\) in favor of \({\bf E}\): -

    - -

    -\[ -\rho = \varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E} -\longrightarrow -W = \frac{\varepsilon_0}{2} \int ({\boldsymbol \nabla} \cdot {\bf E}) ~V d\tau. -\] -

    - -

    -Doing integration by parts and using \({\boldsymbol \nabla} V = -{\bf E}\), -

    - -

    -\[ -W = \frac{\varepsilon_0}{2} \left( \int_{\cal V} E^2 d\tau + \oint_{\cal S} V {\bf E} \cdot d{\bf a} \right) -\label{Gr(2.44)} -\] -

    - -

    -Integrating over all space, -

    -
    -

    - -

    - -

    -\[ - W = \frac{\varepsilon_0}{2} \int E^2 d\tau - \label{eq:Energy_as_int_E2} - \] -

    - -
    - -
    -

    -Example 2.8: Find the energy of a uniformly charged spherical shell of total charge \(q\) and radius \(R\). -

    - -

    -Solution 1: use \ref{Gr(2.43)} in version for surface charges: -

    - -

    -\[ -W = \frac{1}{2} \int \sigma V da. -\] -

    - -

    -The potential at the surface of the sphere is constant, \(V = \frac{1}{4\pi \varepsilon_0} \frac{q}{R}\), so -

    - -

    -\[ -W = \frac{1}{8\pi \varepsilon_0} \frac{q}{R} \int \sigma da = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}. -\] -

    - -

    -Solution 2: use \ref{eq:Energy_as_int_E2}. Inside sphere: \({\bf E} = 0\); outside, -

    - -

    -\[ -{\bf E} = \frac{1}{4\pi \varepsilon_0} q \frac{\hat{\bf r}}{r^2} \rightarrow E^2 = \frac{q^2}{(4\pi \varepsilon_0)^2 r^4}. -\] -

    - -

    -so -\[ - W = \frac{\varepsilon_0}{2} \int_{outside~sphere} \frac{q^2}{(4\pi \varepsilon_0)^2 r^4} r^2 \sin \theta dr d\theta d\phi -= \frac{1}{32 \pi^2 \varepsilon_0} q^2 4\pi \int_R^{\infty} \frac{dr}{r^2} = \frac{1}{8\pi \varepsilon_0} \frac{q^2}{R}. -\] -

    - -
    -
    -
    - - - -
    -
    - -Creative Commons License - -Except where otherwise noted, all content is licensed under a -Creative Commons Attribution 4.0 International License. -
    -
    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    -

    -
    - - -
    - \ No newline at end of file diff --git a/build/ems_es_e_pcd.html b/build/ems_es_e_pcd.html deleted file mode 100644 index 3cd2c6a..0000000 --- a/build/ems_es_e_pcd.html +++ /dev/null @@ -1,1660 +0,0 @@ - - - - - - -Pre-Quantum Electrodynamics - - - - - - - - - -
    -
    -

    -Pre-Quantum Electrodynamics -

    -
    - -
    -
    The Energy of a Point Charge Distribution - - -ems.es.e.pcd
    -
    -

    -The last version can also be rewritten in terms of the potential, -

    - -

    -\[ -W = \frac{1}{2} \sum_{i=1}^m q_i V({\bf r}_i) -\label{Gr(2.42)} -\] -

    -
    -
    - - - - -
    -
    - -Creative Commons License - -Except where otherwise noted, all content is licensed under a -Creative Commons Attribution 4.0 International License. -
    -
    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    -

    -
    - - -
    - \ No newline at end of file diff --git a/build/ems_es_ec.html b/build/ems_es_ec.html index 1b9ebd1..b432d7f 100644 --- a/build/ems_es_ec.html +++ b/build/ems_es_ec.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1652,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ec_b.html b/build/ems_es_ec_b.html index 23b1fdb..b61561f 100644 --- a/build/ems_es_ec_b.html +++ b/build/ems_es_ec_b.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1643,7 +1618,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ec_c.html b/build/ems_es_ec_c.html index 0a444f4..7169073 100644 --- a/build/ems_es_ec_c.html +++ b/build/ems_es_ec_c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1643,7 +1618,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ec_q.html b/build/ems_es_ec_q.html index fc2fe20..c9241a9 100644 --- a/build/ems_es_ec_q.html +++ b/build/ems_es_ec_q.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1643,7 +1618,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ec_s.html b/build/ems_es_ec_s.html index cfbf950..6464498 100644 --- a/build/ems_es_ec_s.html +++ b/build/ems_es_ec_s.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1642,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ef.html b/build/ems_es_ef.html index dde5ab6..fcf1edb 100644 --- a/build/ems_es_ef.html +++ b/build/ems_es_ef.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1652,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ef_Gl.html b/build/ems_es_ef_Gl.html index 40da4cb..cf0c4cb 100644 --- a/build/ems_es_ef_Gl.html +++ b/build/ems_es_ef_Gl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1667,14 +1642,14 @@ sphere of radius \(r\) around the charge,

    so by superposition, we obtain

    -
    +

    -
    +
    • Gr (2.13)
    @@ -1682,7 +1657,7 @@ so by superposition, we obtain
    -
    +

    Gauss' law (in integral form)

    @@ -1714,14 +1689,14 @@ By applying the divergence theorem, and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume is arbitrary, we get

    -
    +

    -
    +
    • Gr (2.14)
    @@ -1729,7 +1704,7 @@ is arbitrary, we get
    -
    +

    Gauss' law in differential form

    @@ -1788,7 +1763,7 @@ cylindrical or plane symmetry. Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.

    -
    +

    Example 2.2: Field outside a uniformly charged sphere of radius \(R\) and total charge \(q\).

    @@ -1819,7 +1794,7 @@ Same as point charge at origin!
    -
    +

    Example 2.3: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.

    @@ -1858,7 +1833,7 @@ Therefore,
    -
    +

    Example 2.4: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).

    @@ -1884,7 +1859,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
    -
    +

    Example 2.5: two infinite planes (put them vertical) carrying equal but opposite uniform surface charge densities \(\pm \sigma\).

    @@ -1912,7 +1887,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ef_cE.html b/build/ems_es_ef_cE.html index 9650066..435e33e 100644 --- a/build/ems_es_ef_cE.html +++ b/build/ems_es_ef_cE.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1626,14 +1601,14 @@ Table of contents Since the work done when moving on a closed path vanishes, we have

    -
    +

    -
    +
    • Gr (2.19)
    @@ -1641,7 +1616,7 @@ Since the work done when moving on a closed path vanishes, we have
    -
    +

    @@ -1659,14 +1634,14 @@ Since the work done when moving on a closed path vanishes, we have which by Stokes' theorem implies that the electrostatic field is curlless,

    -
    +

    -
    +
    • Gr (2.20)
    @@ -1674,12 +1649,16 @@ which by Stokes' theorem implies that
    -
    +
    +

    + +

    +

    \[ - {\boldsymbol \nabla} \times {\bf E} = 0. - \tag{curlE0}\label{curlE0} - \] +{\boldsymbol \nabla} \times {\bf E} = 0. +\tag{curlE0}\label{curlE0} +\]

    @@ -1714,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ef_ccd.html b/build/ems_es_ef_ccd.html index 1ee5c96..98ca3d2 100644 --- a/build/ems_es_ef_ccd.html +++ b/build/ems_es_ef_ccd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1628,14 +1603,14 @@ calculated from Coulomb's law using the superposition principle. Since each inf volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}') d\tau'\), we have

    -
    +

    -
    +
    • Gr4 (2.8)
    @@ -1643,7 +1618,7 @@ volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}')
    -
    +

    @@ -1663,14 +1638,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S} \(\sigma({\bf r})\), we have over an infinitesimal area \(da'\) a charge \(dq' = \sigma({\bf r}') da'\), so

    -
    +

    -
    +
    • Gr4(2.7)
    @@ -1678,7 +1653,7 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
    -
    +

    @@ -1695,14 +1670,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S} Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf r}')\),

    -
    +

    -
    +
    • Gr (2.6)
    @@ -1710,7 +1685,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
    -
    +

    @@ -1724,7 +1699,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
    -
    +

    Example

    @@ -1758,7 +1733,7 @@ most easily by observing that \(\frac{d}{dx} \left( \frac{x}{\sqrt{z^2 + x^2}} \ = \frac{1}{\sqrt{z^2 + x^2}} - \frac{x^2}{(z^2 + x^2)^{3/2}} = \frac{z^2}{(z^2 + x^2)^{3/2}}\), leading to

    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_ef_pc.html b/build/ems_es_ef_pc.html index 74f42c3..de1326b 100644 --- a/build/ems_es_ef_pc.html +++ b/build/ems_es_ef_pc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1636,14 +1611,14 @@ an electric field originating from the source charge. Invoking the superposition principle, we can thus write

    -
    +

    -
    +
    • Gr (2.3)
    @@ -1651,7 +1626,7 @@ Invoking the superposition principle, we can thus write
    -
    +

    @@ -1667,14 +1642,14 @@ Invoking the superposition principle, we can thus write

    with the electric field of a point charge distribution being

    -
    +

    -
    +
    • Gr (2.4)
    @@ -1682,7 +1657,7 @@ with the electric field of a point charge distribution being
    -
    +

    @@ -1700,7 +1675,7 @@ The electric field \({\bf E} ({\bf r})\) is thus the force per unit charge that be exerted if you put a test charge at position \({\bf r}\).

    -
    +

    Example

    @@ -1742,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_efo.html b/build/ems_es_efo.html index 0c7142b..8650427 100644 --- a/build/ems_es_efo.html +++ b/build/ems_es_efo.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1653,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_es_efo_cl.html b/build/ems_es_efo_cl.html index 14fdd82..26da572 100644 --- a/build/ems_es_efo_cl.html +++ b/build/ems_es_efo_cl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1628,14 +1603,14 @@ Table of contents
  • PM 1.4
    • -
    +

    -
    +
    • FLS II (4.9)
    • Gr (2.1)
      • @@ -1647,7 +1622,7 @@ Table of contents
    -
    +

    The force \(F_{t\leftarrow s}\) exerted by a point source charge \(q_s\) sitting at \({\bf r}_s\) on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's law, @@ -1664,7 +1639,7 @@ on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's la

    -

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_dcB_BS.html b/build/ems_ms_dcB_BS.html index 7128ffd..5db05d2 100644 --- a/build/ems_ms_dcB_BS.html +++ b/build/ems_ms_dcB_BS.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1633,7 +1608,7 @@ Apply the divergence: \left(\frac{{\bf J}({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3} \right) \label{Gr(5.46)} \] -Note that we can write (using \ref{Gr(1.101)}) \(\frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} = -{\boldsymbol \nabla} \frac{1}{|{\bf r} - {\bf r}'|}\). Substituting this in and using product rule number 6, +Note that we can write (using div1or) \(\frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} = -{\boldsymbol \nabla} \frac{1}{|{\bf r} - {\bf r}'|}\). Substituting this in and using product rule number 6, \[ {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}), \] @@ -1650,7 +1625,7 @@ we get But \({\bf J}\) depends only on \({\bf r}'\) so \({\boldsymbol \nabla} \times {\bf J} ({\bf r}') = 0\), and since the curl of a gradient always vanishes, we obtain

    -
    +

    \[ {\boldsymbol \nabla} \cdot {\bf B} = 0 @@ -1718,7 +1693,7 @@ at infinity), and in the third step we have used the assumption of steady-state

    We thus obtain in total

    -
    +

    Ampère's law \[ @@ -1735,7 +1710,7 @@ We thus obtain in total \] so

    -
    +

    \[ \oint_{\cal P} {\bf B} \cdot d{\bf l} = \mu_0 I_{enc} \hspace{2cm} @@ -1757,7 +1732,7 @@ Sign ambiguity: resolved by right-hand rule as usual. Ampère's law in magnetostatics takes a parallel role to Gauss's law in electrostatics.

    -
    +

    \paragraph{Example 5.7:} same as Example 5.5, but now with Ampère. \paragraph{Solution:} by symmetry, \({\bf B}\) is circumferential and can only depend on \(s\). Then, @@ -1769,7 +1744,7 @@ choosing an amperian loop at a fixed radius \(s\), we get

    -
    +

    \paragraph{Example 5.8:} uniform surface current \({\bf K} = K \hat{\bf x}\) flowing in \(xy\) plane. \paragraph{Solution:} Biot-Savart: \({\bf B}\) must be perpendicular to \({\bf K}\). Intuition: @@ -1786,7 +1761,7 @@ and along \(\hat{\bf y}\) for \(z < 0\). Amperian loop of width \(l\) punchi

    -
    +

    \paragraph{Example 5.9:} solenoid along \(\hat{\bf z}\), wire carrying current \(I\) doing \(n\) turns per unit length on cylinder of radius \(R\). \paragraph{Solution:} by symmetry, \({\bf B}\) must be along axis of solenoid. Outside: infinitely far away, \({\bf B}\) must vanish. @@ -1807,7 +1782,7 @@ Amperian loop of length \(l\), half-inside and half-outside: i) infinite straight lines, ii) infinite planes, iii) infinite solenoids, iv) toroids.

    -
    +

    \paragraph{Example 5.10:} toroidal coil (no matter the shape, as long as it is rotationally symmetric). \paragraph{Solution:} magnetic field is circumferential everywhere. Outside coil, field again zero. @@ -1838,7 +1813,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_dcB_sc.html b/build/ems_ms_dcB_sc.html index 868059d..17ce9b4 100644 --- a/build/ems_ms_dcB_sc.html +++ b/build/ems_ms_dcB_sc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1652,7 +1627,7 @@ I_{enc} = \int_{\cal S} {\bf J} \cdot d{\bf a} \] Applying Stokes' theorem:

    -
    +

    \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} @@ -1680,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_lf.html b/build/ems_ms_lf.html index 2fd50d1..632d55b 100644 --- a/build/ems_ms_lf.html +++ b/build/ems_ms_lf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    • Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_lf_c.html b/build/ems_ms_lf_c.html index f47982b..ce9a831 100644 --- a/build/ems_ms_lf_c.html +++ b/build/ems_ms_lf_c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1659,7 +1634,7 @@ Since charge is conserved, \] Since this is valid for any volume, we get the

    -
    +

    continuity equation \[ @@ -1684,7 +1659,7 @@ moving at velocity \({\bf v}\) \]

    -
    +

    Forces on wire, surface and volume carrying current densities: \[ @@ -1719,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_lf_pc.html b/build/ems_ms_lf_pc.html index 947fe9a..5fb002c 100644 --- a/build/ems_ms_lf_pc.html +++ b/build/ems_ms_lf_pc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,7 +1600,7 @@ Table of contents

    Force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \({\bf B}\):

    -
    +

    \[ {\bf F}_{mag} = q {\bf v} \times {\bf B} @@ -1637,7 +1612,7 @@ Force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \

    Units of \({\bf B}\): \(N/(A~m)\) is called a {\bf tesla} (symbol: \(T\)). Total electromagnetic force:

    -
    +

    \[ {\bf F}_{mag} = q ({\bf E} + {\bf v} \times {\bf B}) @@ -1647,7 +1622,7 @@ Units of \({\bf B}\): \(N/(A~m)\) is called a {\bf tesla} (symbol: \(T\)). Tot

    -
    +

    \paragraph{Example 5.1:} cyclotron motion. Field \({\bf B}\) pointing into page. Charge \(q > 0\) moves counterclockwise with speed \(v\) on a circle of radius \(R\). Magnetic force points inwards. @@ -1665,7 +1640,7 @@ The {\bf cyclotron frequency} is

    -
    +

    \paragraph{Example 5.2:} cycloid motion. Recommendation: {\it look at it!!}

    @@ -1699,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_vp.html b/build/ems_ms_vp.html index 26b6514..7977832 100644 --- a/build/ems_ms_vp.html +++ b/build/ems_ms_vp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1626,7 +1601,7 @@ Table of contents

    Since \({\boldsymbol \nabla} \cdot {\bf B} = 0\) in magnetostatics, following Helmholtz's theorem we can write

    -
    +

    \[ {\bf B} = {\boldsymbol \nabla} \times {\bf A} @@ -1648,7 +1623,7 @@ add any curlless function (so gradient of a scalar field) to the vector potentia without changing the magnetic field. This is called a {\bf gauge choice} in electrodynamics. For example, we can {\bf always} eliminate the divergence of \({\bf A}\),

    -
    +

    {\bf Example gauge choice:} \[ @@ -1679,7 +1654,7 @@ zero at infinity,

    Under this gauge choice, Ampère's law becomes

    -
    +

    \[ {\boldsymbol \nabla}^2 {\bf A} = -\mu_0 {\bf J} @@ -1692,7 +1667,7 @@ Under this gauge choice, Ampère's law becomes Note: this is a Poisson equation for each component. For currents falling off sufficiently rapidly at infinity,

    -
    +

    \[ {\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int d\tau' \frac{J({\bf r}')}{|{\bf r} - {\bf r}'|} @@ -1704,7 +1679,7 @@ For currents falling off sufficiently rapidly at infinity,

    For line and surface currents, (beware Griffiths' horrendous notation)

    -
    +

    \[ {\bf A}({\bf r}) = \frac{\mu_0}{4\pi} \int dl' \frac{{\bf I ({\bf r}')}}{|{\bf r} - {\bf r}'|}, @@ -1718,7 +1693,7 @@ For line and surface currents, (beware Griffiths' horrendous notation) -

    +

    \paragraph{Example 5.11:} a spherical shell of radius \(R\), carrying a uniform surface charge \(\sigma\), is set spinning at angular velocity \(\omega\). Find the vector potential at \({\bf r}\). @@ -1732,7 +1707,7 @@ the sphere is uniform !

    -
    +

    \paragraph{Example 5.12:} find the vector potential of an infinite solenoid with \(n\) turns pet unit length, radius \(R\) and current \(I\). @@ -1803,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_vp_LC.html b/build/ems_ms_vp_LC.html index 931233d..56e7738 100644 --- a/build/ems_ms_vp_LC.html +++ b/build/ems_ms_vp_LC.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1627,7 +1602,7 @@ Table of contents the magnetic field of a dipole can be written in coordinate-free form. For this: we define a useful object, the

    -
    +

    {\bf Levi-Civita symbol} \[ @@ -1708,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_vp_comp.html b/build/ems_ms_vp_comp.html index 8a1a589..90f3ba6 100644 --- a/build/ems_ms_vp_comp.html +++ b/build/ems_ms_vp_comp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1653,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    • Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_vp_mbc.html b/build/ems_ms_vp_mbc.html index 2013958..5adc52f 100644 --- a/build/ems_ms_vp_mbc.html +++ b/build/ems_ms_vp_mbc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ B^{\parallel}_{above} - B^{\parallel}_{below} = \mu_0 K So: component of \({\bf B}\) that is parallel to surface but perpendicular to current flow is discontinuous, whereas the one parallel to the flow is continuous. In vector notation:

    -
    +

    \[ {\bf B}_{above} - {\bf B}_{below} = \mu_0 {\bf K} \times \hat{\bf n}, @@ -1658,7 +1633,7 @@ is discontinuous, whereas the one parallel to the flow is continuous. In vector

    where \(\hat{\bf n}\) points upwards. For the vector potential, the relations are

    -
    +

    \[ {\bf A}_{above} = {\bf A}_{below} @@ -1704,7 +1679,7 @@ What is left is \] Therefore, reidentifying the normal component, we get

    -
    +

    \[ \frac{\partial {\bf A}_{above}}{\partial n} - \frac{\partial {\bf A}_{below}}{\partial n} @@ -1732,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/ems_ms_vp_me.html b/build/ems_ms_vp_me.html index 425d2f1..74541ae 100644 --- a/build/ems_ms_vp_me.html +++ b/build/ems_ms_vp_me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1677,7 +1652,7 @@ Using equation \ref{Gr(1.108)} from Problem 1.61, with \({\bf c} = \hat {\bf r}\),

    -
    +

    \paragraph{Parenthesis: Problem 1.61 (e)} Start from Stokes' theorem, \[ @@ -1719,7 +1694,7 @@ Back to our problem: \] and defining the

    -
    +

    magnetic dipole moment \[ @@ -1732,7 +1707,7 @@ and defining the

    we obtain the convenient expression for the

    -
    +

    {\bf dipole term of the vector potential} \[ @@ -1743,7 +1718,7 @@ we obtain the convenient expression for the

    -
    +

    \paragraph{Example 5.13:} find magnetic dipole moment of bookend shape of Gr. Fig. 5.52. All sides have length \(w\) and carry current \(I\). @@ -1788,7 +1763,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm.html b/build/emsm.html index 098fbe8..491189b 100644 --- a/build/emsm.html +++ b/build/emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emsm
    -
    - +
    + Prerequisites
      @@ -1633,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1671,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm.html b/build/emsm_esm.html index fb3a908..e8da694 100644 --- a/build/emsm_esm.html +++ b/build/emsm_esm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1648,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_D.html b/build/emsm_esm_D.html index 117148e..1e39cd2 100644 --- a/build/emsm_esm_D.html +++ b/build/emsm_esm_D.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1626,7 +1601,7 @@ Table of contents

    Field due to polarization: effectively comes from bound surface and volume charges,

    -
    +

    \[ \rho_b = -{\boldsymbol \nabla} \cdot {\bf P}, \hspace{1cm} \sigma_b = {\bf P} \cdot \hat{\bf n}. @@ -1655,7 +1630,7 @@ Convenient way of writing: \] Defining the

    -
    +

    Electric displacement \({\bf D}\) \[ @@ -1668,7 +1643,7 @@ Defining the

    Gauss's law becomes

    -
    +

    \[ {\boldsymbol \nabla} \cdot {\bf D} = \rho_f @@ -1680,7 +1655,7 @@ Gauss's law becomes

    or in integral form

    -
    +

    \[ \oint d{\bf a} \cdot {\bf D} = Q_{f_{enc}} @@ -1690,7 +1665,7 @@ or in integral form

    -
    +

    \paragraph{Example 4.4:} long straight wire, uniform line charge \(\lambda\), surrounded by rubber insulation to radius \(a\). Find \({\bf D}\). @@ -1749,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_D_bc.html b/build/emsm_esm_D_bc.html index f9b7343..e50a016 100644 --- a/build/emsm_esm_D_bc.html +++ b/build/emsm_esm_D_bc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1630,7 +1605,7 @@ D^{\perp}_{above} - D^{\perp}_{below} = \sigma_f \] Then, \ref{Gr(4.25)} give

    -
    +

    \[ {\bf D}^{\parallel}_{above} - {\bf D}^{\parallel}_{below} = {\bf P}^{\parallel}_{above} - {\bf P}^{\parallel}_{below}. @@ -1656,7 +1631,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_di.html b/build/emsm_esm_di.html index 2300d34..699c4cb 100644 --- a/build/emsm_esm_di.html +++ b/build/emsm_esm_di.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_di_ld.html b/build/emsm_esm_di_ld.html index f111b7f..f8090fe 100644 --- a/build/emsm_esm_di_ld.html +++ b/build/emsm_esm_di_ld.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1630,7 +1605,7 @@ Table of contents

    For many substances: polarization is proportional to field, if the latter isn't too strong:

    -
    +

    \[ {\bf P} = \varepsilon_0 \chi_e {\bf E} @@ -1660,7 +1635,7 @@ In linear dielectrics: \] so

    -
    +

    \[ {\bf D} = \varepsilon {\bf E} @@ -1682,7 +1657,7 @@ This is all just nomenclature, everything is already in \ref{Gr(4.30)}. -

    +

    \paragraph{Example 4.5:} metal sphere of radius \(a\) carrying charge \(Q\), surrounded out to radius \(b\) by a linear dielectric material of permittivity \(\varepsilon\). Find potential at center (relative to infinity). @@ -1738,7 +1713,7 @@ of {\it e.g.} \({\bf P}\) would not vanish. Only case where parallel works: space entirely filled with homogeneous linear dielectric.

    -
    +

    \paragraph{Example 4.6:} parallel-plate capacitor filled with insulating material of dielectric constant \(\varepsilon_r\). What is the effect on the capacitance ? @@ -1773,7 +1748,7 @@ If \(\rho = 0\), any net charge is on surface, potential then obeys Laplace.

    Convenient to rewrite boundary conditions in terms of free charge: from \ref{Gr(4.26)},

    -
    +

    \[ \varepsilon_{above} E^{\perp}_{above} - \varepsilon_{below} E^{\perp}_{below} = \sigma_f @@ -1785,7 +1760,7 @@ Convenient to rewrite boundary conditions in terms of free charge: from \ref{Gr

    or in terms of the potential,

    -
    +

    \[ \varepsilon_{above} \frac{\partial V_{above}}{\partial n} - @@ -1798,7 +1773,7 @@ or in terms of the potential,

    Potential itself is continuous,

    -
    +

    \[ V_{above} = V_{below} @@ -1810,7 +1785,7 @@ Potential itself is continuous, -

    +

    \paragraph{Example 4.7:} sphere of homogeneous dielectric material in uniform electric field \({\bf E}_0\). Find electric field inside sphere. @@ -1868,7 +1843,7 @@ Thus, -

    +

    \paragraph{Example 4.8:} suppose region below \(z = 0\) is filled with uniform linear dielectric with susceptibility \(\chi_e\). Calculate force on point charge \(q\) situated a distance \(d\) above origin. @@ -1972,7 +1947,7 @@ Special case of linear isotropic dielectric: \({\bf D} = \varepsilon {\bf E}\), \] Total work done:

    -
    +

    \[ W = \frac{1}{2} \int d\tau {\bf D} \cdot {\bf E} @@ -2074,7 +2049,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_fpo.html b/build/emsm_esm_fpo.html index e65d0bc..c92ea5c 100644 --- a/build/emsm_esm_fpo.html +++ b/build/emsm_esm_fpo.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1666,7 +1641,7 @@ V({\bf r}) = \frac{1}{4\pi\varepsilon_0} \oint_{\cal S} d{\bf a}' ⋅ \frac{{\bf \] Interpretation: first terms is like contribution of a surface charge,

    -
    +

    \[ \sigma_b({\bf r}) = {\bf P} ({\bf r}) \cdot \hat{\bf n} @@ -1678,7 +1653,7 @@ Interpretation: first terms is like contribution of a surface charge,

    and second term looks like contribution of a volume charge,

    -
    +

    \[ \rho_b ({\bf r}) = -{\boldsymbol \nabla} \cdot {\bf P} ({\bf r}) @@ -1690,7 +1665,7 @@ and second term looks like contribution of a volume charge,

    Using these definitions,

    -
    +

    \[ V({\bf r}) = \frac{1}{4\pi\varepsilon_0} \oint_{\cal S} d{\bf a}' ⋅ \frac{σ_b ({\bf r}')}{|{\bf r} - {\bf r}'|} @@ -1709,7 +1684,7 @@ These {\bf bound charges} faithfully represent the object's sources for electric

    -
    +

    \paragraph{Example 4.2:} electric field produced by uniformly polarized sphere of radius \(R\). \paragraph{Solution:} put \(z\) axis along \({\bf P}\). Since \({\bf P}\) is uniform, \(\rho_b = 0\). @@ -1766,7 +1741,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_fpo_fid.html b/build/emsm_esm_fpo_fid.html index dadaf3c..233b86f 100644 --- a/build/emsm_esm_fpo_fid.html +++ b/build/emsm_esm_fpo_fid.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1654,7 +1629,7 @@ this is given by \ref{Gr(3.105)} (see Example below) irrespective of charge distribution within sphere.

    -
    +

    \paragraph{Problem 3.47:} show that the average field inside sphere of radius \(R\) due to all charges within the sphere is @@ -1760,7 +1735,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_fpo_pibc.html b/build/emsm_esm_fpo_pibc.html index 13fdd22..ff5b59f 100644 --- a/build/emsm_esm_fpo_pibc.html +++ b/build/emsm_esm_fpo_pibc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1663,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_p.html b/build/emsm_esm_p.html index 7ffacf4..af25afd 100644 --- a/build/emsm_esm_p.html +++ b/build/emsm_esm_p.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1704,7 +1679,7 @@ dipole moment pointing from \(O^-\) along bisector.

    Torque on dipole: if field is uniform, overall force on dipole cancels, but torque remains:

    -
    +

    {\bf Torque on a dipole:} \[ @@ -1721,7 +1696,7 @@ If field is non-uniform, \] so

    -
    +

    \[ {\bf F} = ({\bf p} \cdot {\boldsymbol \nabla}) {\bf E} @@ -1776,7 +1751,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_esm_s.html b/build/emsm_esm_s.html index e190fc1..58318c9 100644 --- a/build/emsm_esm_s.html +++ b/build/emsm_esm_s.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1640,7 +1615,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm.html b/build/emsm_msm.html index 634c8fe..246c75c 100644 --- a/build/emsm_msm.html +++ b/build/emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1623,8 +1598,8 @@ Table of contents emsm.msm
    -
    - +
    + Prerequisites
      @@ -1633,8 +1608,8 @@ Prerequisites
    -
    - +
    + Objectives
      @@ -1671,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_H.html b/build/emsm_msm_H.html index afebd5b..955358f 100644 --- a/build/emsm_msm_H.html +++ b/build/emsm_msm_H.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1644,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_H_A.html b/build/emsm_msm_H_A.html index 685ea08..ccabb47 100644 --- a/build/emsm_msm_H_A.html +++ b/build/emsm_msm_H_A.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1636,7 +1611,7 @@ Ampère's law: \] so we can define

    -
    +

    \[ {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M} @@ -1648,7 +1623,7 @@ so we can define

    and rewrite Ampère's law as

    -
    +

    \[ {\boldsymbol \nabla} \times {\bf H} = {\bf J}_f @@ -1660,7 +1635,7 @@ and rewrite Ampère's law as

    or in integral form,

    -
    +

    \[ \oint {\bf H} \cdot d{\bf l} = I_{f_{enc}} @@ -1674,7 +1649,7 @@ or in integral form, rewrite Ampère's law in terms of free currents alone. Bound currents come along for the ride.

    -
    +

    \paragraph{Example 6.2:} long copper rod radius \({\bf R}\) carries uniformly distributed free current \(I\). Find \({\bf H}\) inside and outside rod. @@ -1791,7 +1766,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_a.html b/build/emsm_msm_a.html index d50995f..d858c18 100644 --- a/build/emsm_msm_a.html +++ b/build/emsm_msm_a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1682,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_fmo.html b/build/emsm_msm_fmo.html index 1e4b7bc..83debbe 100644 --- a/build/emsm_msm_fmo.html +++ b/build/emsm_msm_fmo.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_fmo_bc.html b/build/emsm_msm_fmo_bc.html index 6bafdd3..af31d1f 100644 --- a/build/emsm_msm_fmo_bc.html +++ b/build/emsm_msm_fmo_bc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1633,7 +1608,7 @@ For a single dipole: refer to \ref{Gr(5.83)} (vector potential of single dipole For a chunk of material with local magnetization \({\bf M} ({\bf r})\), by the principle of superposition we thus have:

    -
    +

    \[ {\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int_{\cal V} d\tau' ~\frac{{\bf M} ({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3} @@ -1675,7 +1650,7 @@ Problem 1.61 b) (p.56): leads to \] Reinterpretation: first term: potential from volume current,

    -
    +

    \[ {\bf J}_b = {\boldsymbol \nabla} \times {\bf M} @@ -1687,7 +1662,7 @@ Reinterpretation: first term: potential from volume current,

    second term: potential from surface current,

    -
    +

    \[ {\bf K}_b = {\bf M} \times \hat{\bf n} @@ -1699,7 +1674,7 @@ second term: potential from surface current,

    With these definitions,

    -
    +

    \[ {\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} ∫_{\cal V} dτ' \frac{{\bf J}_b ({\bf r}')}{|{\bf r} - {\bf r}'|} @@ -1720,7 +1695,7 @@ in the volume and surface of the material. -

    +

    \paragraph{Example 6.1:} find field of uniformly magnetized sphere. \paragraph{Solution:} put z axis along \({\bf M}\). @@ -1764,7 +1739,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_fmo_fim.html b/build/emsm_msm_fmo_fim.html index d40687f..1c1b265 100644 --- a/build/emsm_msm_fmo_fim.html +++ b/build/emsm_msm_fmo_fim.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1642,7 +1617,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_fmo_pibc.html b/build/emsm_msm_fmo_pibc.html index 2244be4..37cfd5b 100644 --- a/build/emsm_msm_fmo_pibc.html +++ b/build/emsm_msm_fmo_pibc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_lnlm.html b/build/emsm_msm_lnlm.html index 6ed9d3a..08a25e5 100644 --- a/build/emsm_msm_lnlm.html +++ b/build/emsm_msm_lnlm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_lnlm_fm.html b/build/emsm_msm_lnlm_fm.html index ecfe6f8..f7afb7d 100644 --- a/build/emsm_msm_lnlm_fm.html +++ b/build/emsm_msm_lnlm_fm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1643,7 +1618,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_lnlm_sp.html b/build/emsm_msm_lnlm_sp.html index d8cb12e..968d1ad 100644 --- a/build/emsm_msm_lnlm_sp.html +++ b/build/emsm_msm_lnlm_sp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1627,7 +1602,7 @@ In para/diamagnets: when \({\bf B}\) is removed, \({\bf M}\) disappears. For n fields, proportionality. Custom (slightly different than for dielectrics): {\bf magnetic susceptibility} \(\chi_m\) defined as

    -
    +

    \[ {\bf M} = \chi_m {\bf H} @@ -1683,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_m.html b/build/emsm_msm_m.html index 6ccccb9..6057a39 100644 --- a/build/emsm_msm_m.html +++ b/build/emsm_msm_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1646,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_m_dpf.html b/build/emsm_msm_m_dpf.html index 584fe23..a04c5b0 100644 --- a/build/emsm_msm_m_dpf.html +++ b/build/emsm_msm_m_dpf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1633,8 +1608,8 @@ Table of contents

    - @@ -1656,7 +1631,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/emsm_msm_m_fdi.html b/build/emsm_msm_m_fdi.html index 96b216d..854ee96 100644 --- a/build/emsm_msm_m_fdi.html +++ b/build/emsm_msm_m_fdi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1625,7 +1600,7 @@ Table of contents

    Torque on magnetic dipole:

    -
    +

    \[ {\bf N} = {\bf m} \times {\bf B} @@ -1643,7 +1618,7 @@ Similar to electric analogue, \ref{Gr(4.4)}, \({\bf N} = {\bf p} \times {\bf E}\

    In a non-uniform field: (to be done in Problem 6.4):

    -
    +

    \[ {\bf F} = {\boldsymbol \nabla} ({\bf m} \cdot {\bf B}) @@ -1670,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/in.html b/build/in.html index 156bbbe..c751a60 100644 --- a/build/in.html +++ b/build/in.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/in_p.html b/build/in_p.html index 56a1c80..ce54f6a 100644 --- a/build/in_p.html +++ b/build/in_p.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1668,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/in_t.html b/build/in_t.html index 95ab20b..33490c7 100644 --- a/build/in_t.html +++ b/build/in_t.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1645,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/in_t_c.html b/build/in_t_c.html index 626d54c..81e9ff2 100644 --- a/build/in_t_c.html +++ b/build/in_t_c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1627,8 +1602,8 @@ Throughout these lecture notes, boxes with contextual colors are used, serving specific purposes. These are:

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    - +
    + Prerequisites

    @@ -1641,8 +1616,8 @@ Specific prerequisites for understanding the upcoming material

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    - +
    + Objectives

    @@ -1655,8 +1630,8 @@ Objectives for this part: what you should learn by reading this

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    - +
    + Core

    @@ -1664,8 +1639,8 @@ Core material: you have to know this by heart, and how to use it.

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    - +
    + Main

    @@ -1673,8 +1648,8 @@ Main matter which you should know how to use.

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    - +
    + Example

    @@ -1682,8 +1657,8 @@ Example of the concepts just covered.

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    - +
    + Info

    @@ -1691,8 +1666,8 @@ Additional (contextual) information.

    -
    - +
    + Historical Context

    @@ -1716,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

    -

    Created: 2022-02-09 Wed 07:31

    +

    Created: 2022-02-09 Wed 22:40

    diff --git a/build/in_t_l.html b/build/in_t_l.html index feefa26..3688979 100644 --- a/build/in_t_l.html +++ b/build/in_t_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
    • -
  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
  • @@ -430,29 +426,8 @@ Table of contents
  • - -
    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1680,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

    Author: Jean-Sébastien Caux

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    Created: 2022-02-09 Wed 07:31

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    Created: 2022-02-09 Wed 22:40

    diff --git a/build/index.html b/build/index.html index e47c0c8..5e9bf55 100644 --- a/build/index.html +++ b/build/index.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -408,17 +408,13 @@ Table of contents
  • Field in terms of the potentialems.es.ep.fp -
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  • -Comments on the Electrostatic Potentialems.es.ep.c -
  • Example calculations for the potentialems.es.ep.ex
  • -The Poisson Equation and the Laplace Equationems.es.ep.PL +Poisson's and Laplace's Equationsems.es.ep.PL
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    - Electrostatic Energy from the Potentialems.es.e - - - -
  • @@ -1676,20 +1651,12 @@ Table of contents
    • -
  • -Electrostatic Energy from the Potentialems.es.e - -
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  • Electrostatic Energy from the Potentialems.es.e
  • Conductorsems.es.c