diff --git a/build/a.html b/build/a.html index e616704..0649223 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/a_l.html b/build/a_l.html index 3b961f2..1b20b73 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1728,7 +1728,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c.html b/build/c.html index 6bc5b23..51077b5 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m.html b/build/c_m.html index 8703dbd..8dfbb61 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index 92c8f2f..0ec2efc 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index c19e2c6..627bdf0 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1667,14 +1667,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Gradient
-
+

-
+
  • Gr4(1.79)
@@ -1695,14 +1695,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Divergence
-
+

-
+
  • Gr4(2.21)
@@ -1723,14 +1723,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Curl
-
+

-
+
  • Gr4(2.21)
@@ -1779,7 +1779,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index f08d277..24d66c1 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 31e58ba..4b25f18 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1631,14 +1631,14 @@ which \(r\) is the distance from the chosen origin, The usual Cartesian coordinates relate to spherical coordinates according to

-
+

-
+
@@ -1663,14 +1663,14 @@ A generic vector can be expressed as where the explicit relation between spherical and Cartesian unit vectors is

-
+

-
+
@@ -1693,14 +1693,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).

An infinitesimal displacement \(d{\bf l}\) can be written as

-
+

-
+
@@ -1716,14 +1716,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin

Infinitesimal volume element:

-
+

-
+
@@ -1744,14 +1744,14 @@ Infinitesimal surface element: depends on situation.
Gradient
-
+

-
+
@@ -1768,14 +1768,14 @@ Infinitesimal surface element: depends on situation.
Divergence
-
+

-
+
@@ -1792,14 +1792,14 @@ Infinitesimal surface element: depends on situation.
Curl
-
+

-
+
@@ -1818,14 +1818,14 @@ Infinitesimal surface element: depends on situation.
Laplacian
-
+

-
+
@@ -1859,7 +1859,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index bf78727..e17bf29 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index 1d7070d..9be1730 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index 111f7c7..548b752 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,9 +1623,9 @@ 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 @@ -1634,44 +1634,44 @@ 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
+
+

-
+
@@ -1702,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 9670f88..052755a 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index 929dcd3..474b607 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 230639f..aa02c9f 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1672,7 +1672,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 9754735..e81501f 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1630,14 +1630,14 @@ explicited as follows:

Gradient of a product:

-
+

-
+
  • Gr (3)
  • W (1-111)
    • @@ -1657,14 +1657,14 @@ explicited as follows:

    Gradient of a scalar product:

    -
    +

    -
    +
    • Gr (4)
    • W (1-112)
      • @@ -1684,14 +1684,14 @@ explicited as follows:

      Divergence of a product:

      -
      +

      -
      +
      • Gr (5)
      • W (1-115)
        • @@ -1711,14 +1711,14 @@ explicited as follows:

        Divergence of a cross product:

        -
        +

        -
        +
        • Gr (6)
        • W (1-116)
          • @@ -1738,14 +1738,14 @@ explicited as follows:

          Curl of a product:

          -
          +

          -
          +
          • Gr (7)
          • W (1-118)
            • @@ -1765,14 +1765,14 @@ explicited as follows:

            Curl of a cross product:

            -
            +

            -
            +
            • Gr (8)
            • W (1-119)
              • @@ -1813,7 +1813,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_dd.html b/build/c_m_dd.html index d0f9c15..4b7c36d 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index 26fce02..7a47f7d 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index c4ecb4f..7856927 100644 --- a/build/c_m_dd_3d.html +++ b/build/c_m_dd_3d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1644,14 +1644,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox: More generally,

            -
            +

            -
            +
            • Gr (1.100)
            @@ -1670,14 +1670,14 @@ More generally, Since

            -
            +

            -
            +
            • Gr (1.101)
            @@ -1693,14 +1693,14 @@ Since

            we have that

            -
            +

            -
            +
            • Gr (1.102)
            @@ -1732,7 +1732,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index ac4d8e3..bf2eca5 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic.html b/build/c_m_ic.html index b0d8c0c..6fb9c66 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index 3a06c5d..0880cb2 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index 0d8ae01..886077c 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index 9f37cfe..acc38b9 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index 2d66d11..144a4da 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1658,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 7de0157..3e3c9b0 100644 --- a/build/c_m_ic_lsv.html +++ b/build/c_m_ic_lsv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,9 +1623,9 @@ Table of contents
            -
            -
            Line Integrals
            -
            +
            +
            Line Integrals
            +

            \[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} @@ -1654,9 +1654,9 @@ Integral over a closed loop:

            -
            -
            Surface Integrals
            -
            +
            +
            Surface Integrals
            +

            \[ \int_{\cal S} {\bf v} \cdot d{\bf a} @@ -1676,9 +1676,9 @@ Over a closed surface:

            -
            -
            Volume Integrals
            -
            +
            +
            Volume Integrals
            +

            \[ \int_{\cal V} T d\tau @@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index df7178c..efb3a98 100644 --- a/build/c_m_ic_stokes.html +++ b/build/c_m_ic_stokes.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,14 +1621,14 @@ Table of contents c.m.ic.stokes
            -
            +

            -
            +
            • Gr (1.57)
            @@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_uf.html b/build/c_m_uf.html index b48672a..6a2426a 100644 --- a/build/c_m_uf.html +++ b/build/c_m_uf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 2072857..0330234 100644 --- a/build/c_m_uf_cyl.html +++ b/build/c_m_uf_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index b9ae5c5..e7bacff 100644 --- a/build/c_m_uf_sph.html +++ b/build/c_m_uf_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index fd15da7..6ffe56f 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va.html b/build/c_m_va.html index 883aaad..b7de0c2 100644 --- a/build/c_m_va.html +++ b/build/c_m_va.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index b607b97..2f32b27 100644 --- a/build/c_m_va_cp.html +++ b/build/c_m_va_cp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1678,7 +1678,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index 9026ca3..f51c9e7 100644 --- a/build/c_m_va_n.html +++ b/build/c_m_va_n.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index fb80325..8c49e37 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1707,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index 1c54b44..df3f07f 100644 --- a/build/c_m_va_sp.html +++ b/build/c_m_va_sp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index 0148ef2..d7bd155 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_vf.html b/build/c_m_vf.html index eb3e223..b55e604 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index d85d914..ecf3737 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index 4675dd8..c027a39 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1692,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

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            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d.html b/build/d.html index 1b6b8b8..d2ca20b 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,8 +1622,8 @@ Table of contents d
            -
            - +
            + Objectives @@ -1687,7 +1687,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d_emd.html b/build/d_emd.html index 7b226fc..fbc019f 100644 --- a/build/d_emd.html +++ b/build/d_emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1667,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index a00a8e6..f97d647 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

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            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index 282d57e..045efcc 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d_emf.html b/build/d_emf.html index dc5a69e..12f9c51 100644 --- a/build/d_emf.html +++ b/build/d_emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-02-21 Mon 20:41

            +

            Created: 2022-03-01 Tue 08:14

            diff --git a/build/d_ems.html b/build/d_ems.html index bb68c8c..4e95169 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1684,7 +1684,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-02-21 Mon 20:41

          +

          Created: 2022-03-01 Tue 08:14

          diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index d7badb4..3069b1b 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1667,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-02-21 Mon 20:41

          +

          Created: 2022-03-01 Tue 08:14

          diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 1ec8260..47e909a 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-02-21 Mon 20:41

          +

          Created: 2022-03-01 Tue 08:14

          diff --git a/build/d_emsm.html b/build/d_emsm.html index 9dfa08d..3f23763 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1667,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-02-21 Mon 20:41

          +

          Created: 2022-03-01 Tue 08:14

          diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index 589f24f..e6fecba 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/d_m.html b/build/d_m.html index 234d741..301780e 100644 --- a/build/d_m.html +++ b/build/d_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/d_red.html b/build/d_red.html index 0dd0bdc..c214200 100644 --- a/build/d_red.html +++ b/build/d_red.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1669,7 +1669,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd.html b/build/emd.html index 818c184..c2f9e6f 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,8 +1622,8 @@ Table of contents emd
        -
        - +
        + Prerequisites
          @@ -1632,8 +1632,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 0f61f2e..7b50334 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index 749c0c2..40f2617 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1662,7 +1662,7 @@ 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), \[ @@ -1684,7 +1684,7 @@ to the rate of change of the magnetic flux, \] so we obtain

        -
        +

        Faraday's law (integral form N.B.: for a stationary loop) \[ @@ -1702,7 +1702,7 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

        -
        +

        Faraday's law (differential form) \[ @@ -1739,7 +1739,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index 0233365..750df8a 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1646,9 +1646,26 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo \label{Gr(7.30)} \] Generalization to volume currents: +

        +
        +

        + + + + +

        +
        +
          +
        • gr (7.31)
          • +
        + +
        + +
        +

        \[ W = \frac{1}{2} \int_{\cal V} ({\bf A} \cdot {\bf J}) d\tau -\label{Gr(7.31)} +\tag{W_intAJ}\label{W_intAJ} \] Even better: use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\): \[ @@ -1681,7 +1698,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold \] 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 @@ -1702,7 +1719,7 @@ 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 \(l\). @@ -1741,7 +1758,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 43e2876..261d578 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1661,7 +1661,7 @@ M_{12} = M_{21} \]

        -
        +

        \paragraph{Example 7.10:} short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside @@ -1711,7 +1711,7 @@ 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 \(h\)) @@ -1738,7 +1738,7 @@ 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 ? @@ -1777,7 +1777,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index e20eb7f..dd2af9b 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,7 +1645,7 @@ law in integral form: -
        +

        {\bf Example 7.7:} \({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? @@ -1661,7 +1661,7 @@ 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 \(a < b\), @@ -1695,7 +1695,7 @@ 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. @@ -1743,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Me.html b/build/emd_Me.html index 26ca2e4..72450b8 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index d91dc74..e552283 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ Table of contents

        Full set of equations for the electromagnetic field:

        -
        +

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

        @@ -1640,7 +1640,7 @@ Full set of equations for the electromagnetic field:

        Complement:

        -
        +

        {\bf Force law} \[ @@ -1664,7 +1664,7 @@ 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, \\ @@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index e358948..9d6d1ff 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1631,7 +1631,7 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

        -
        +

        \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} @@ -1655,7 +1655,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

        Maxwell baptized this term the

        -
        +

        {\bf Displacement current} \[ @@ -1703,7 +1703,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index 0e44810..88f3738 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index fb2c4f5..499a8a7 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1671,7 +1671,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce.html b/build/emd_ce.html index efc5fff..9075b9d 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,8 +1622,8 @@ Table of contents emd.ce
        -
        - +
        + Prerequisites
          @@ -1631,8 +1631,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1670,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 55a1f40..04223ec 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ Table of contents

        The angular momentum of EM fields is directly given by

        -
        +

        {\bf Angular momentum of EM fields} \[ @@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index c3f8e84..befd23a 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

        -
        +

        {\bf Continuity equation} \[ @@ -1684,7 +1684,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index ff7694a..3995e55 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,7 +1636,7 @@ 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} \[ @@ -1648,7 +1648,7 @@ 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} \[ @@ -1677,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index f035020..fcb15eb 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1678,7 +1678,7 @@ and similarly for \({\boldsymbol B}\). We thus get

        This expression can be greatly simplified by introducing the

        -
        +

        {\bf Maxwell stress tensor} \[ @@ -1701,7 +1701,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction

        We then obtain

        -
        +

        {\bf EM force per unit volume} \[ @@ -1713,7 +1713,7 @@ We then obtain

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

        -
        +

        {\bf Total force on charges in volume} \[ @@ -1742,7 +1742,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index ee19a9d..fbfea9b 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1690,7 +1690,7 @@ so we get Substituting this in \ref{Gr(8.6)} and using the divergence theorem, we obtain

        -
        +

        {\bf Poynting's theorem} \[ @@ -1715,7 +1715,7 @@ 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} \[ @@ -1728,7 +1728,7 @@ Energy per unit time, per unit area carried by EM fields:

        We can thus express Poynting's theorem more compactly:

        -
        +

        {\bf Poynting's theorem} \[ @@ -1741,7 +1741,7 @@ We can thus express Poynting's theorem more compactly:

        where we have defined the total

        -
        +

        {\bf Energy in electromagnetic fields} \[ @@ -1764,7 +1764,7 @@ Then, \] so we get the

        -
        +

        {\bf Poynting theorem (differential form)} \[ @@ -1781,7 +1781,7 @@ 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 @@ -1826,7 +1826,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_emw.html b/build/emd_emw.html index 420337f..3d28bc1 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,8 +1622,8 @@ Table of contents emd.emw
        -
        - +
        + Prerequisites
          @@ -1632,8 +1632,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index 8b21f34..3fc6cb3 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1652,7 +1652,7 @@ so for a monochromatic EM plan wave, \] or more succinctly:

        -
        +

        {\bf Poynting vector of a monochromatic EM wave} \[ @@ -1668,7 +1668,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

        Similary, we get the

        -
        +

        {\bf Momentum density of a monochromatic EM wave} \[ @@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index 9803efd..474d7a1 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1653,7 +1653,7 @@ 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} \[ @@ -1697,7 +1697,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index 18935fa..f21e8ac 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1650,7 +1650,7 @@ 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} \[ @@ -1706,7 +1706,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

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        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emdm.html b/build/emdm.html index d387e64..51782fb 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 4468f41..7eb9d64 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-02-21 Mon 20:41

        +

        Created: 2022-03-01 Tue 08:14

        diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 87647c0..171e7b0 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1657,7 +1657,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

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        {\bf Polarization current density} \[ @@ -1675,7 +1675,7 @@ 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:

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      + Prerequisites
        @@ -1635,8 +1635,8 @@ Prerequisites
      -
      - +
      + Objectives
        @@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

      -

      Created: 2022-02-21 Mon 20:41

      +

      Created: 2022-03-01 Tue 08:14

      diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html index 6935e66..2ba05ad 100644 --- a/build/ems_ca_fe.html +++ b/build/ems_ca_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1632,7 +1632,7 @@ 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

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      @@ -1646,7 +1646,7 @@ from calculating either the field according to p_vcd

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      @@ -1666,7 +1666,7 @@ condition curlE0 can be expressed as th 🐟

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      @@ -1682,7 +1682,7 @@ condition curlE0 can be expressed as th

      In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation Lap

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      @@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

      -

      Created: 2022-02-21 Mon 20:41

      +

      Created: 2022-03-01 Tue 08:14

      diff --git a/build/ems_ca_fe_L.html b/build/ems_ca_fe_L.html index 0ec2ddc..f7ecf48 100644 --- a/build/ems_ca_fe_L.html +++ b/build/ems_ca_fe_L.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,14 +1638,14 @@ In one dimension, the potential is a single-variable function \(\phi (x)\) and the Laplace equation reads

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      @@ -1660,14 +1660,14 @@ function \(\phi (x)\) and the Laplace equation reads

      The solution to this is

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      +

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      • Gr (3.6)
      @@ -1726,14 +1726,14 @@ In two dimensions, the potential becomes a function of two variables (here: \(x\) and \(y\)), so Laplace's equation now reads

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      @@ -1786,14 +1786,14 @@ a point equals its value averaged over a sphere \(S_R({\bf r})\) of any radius \(R\) centered on this point (and of course not containing any charges),

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      @@ -1805,8 +1805,8 @@ a point equals its value averaged over a sphere \]

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      + Physicist's proof

      @@ -1868,8 +1868,8 @@ proving the theorem.

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      + Formal proof @@ -1919,14 +1919,14 @@ we get the following general

      Theorem:

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      @@ -1979,19 +1979,19 @@ 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.

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      -
      +

      Earnshaw's theorem (physical version)

      @@ -2110,7 +2110,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

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      Created: 2022-02-21 Mon 20:41

      +

      Created: 2022-03-01 Tue 08:14

      diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html index 88b2762..bd863e5 100644 --- a/build/ems_ca_fe_g.html +++ b/build/ems_ca_fe_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,11 +1621,11 @@ Table of contents ems.ca.fe.g
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      George Green

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    +

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    • FLS II (4.9)
    • Gr (2.1)
      • @@ -1646,7 +1646,7 @@ Table of contents
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    +

    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, @@ -1663,7 +1663,7 @@ on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's la

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Electrical forces obey the principle of superposition: the electrostatic interaction between two charged particles is left entirely @@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/ems_es_ep.html b/build/ems_es_ep.html index 2a0c06f..0b4b99e 100644 --- a/build/ems_es_ep.html +++ b/build/ems_es_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1654,7 +1654,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/ems_es_ep_PL.html b/build/ems_es_ep_PL.html index 7486037..3deda56 100644 --- a/build/ems_es_ep_PL.html +++ b/build/ems_es_ep_PL.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1630,14 +1630,14 @@ Our two fundamental equations for the electrostatic field are \] For the electrostatic potential, by Emgp, Gauss's law becomes

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+

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  • Gr (2.24)
  • PM (2.72)
    • @@ -1646,7 +1646,7 @@ For the electrostatic potential, by Emgp,
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+

Poisson's equation \[ @@ -1659,14 +1659,14 @@ For the electrostatic potential, by Emgp,

When the charge density vanishes, it becomes more simply

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+

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  • Gr (2.25)
  • PM (2.75)
    • @@ -1675,7 +1675,7 @@ When the charge density vanishes, it becomes more simply
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+

Laplace's Equation \[ @@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/ems_es_ep_bc.html b/build/ems_es_ep_bc.html index ab19cc0..feec1a6 100644 --- a/build/ems_es_ep_bc.html +++ b/build/ems_es_ep_bc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1658,14 +1658,14 @@ applied to a small loop straddling the surface:

We can unify both boundary conditions into a single equation:

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+

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  • Gr (2.33)
@@ -1700,14 +1700,14 @@ The gradient however inherits the discontinuity of the electrostatic field, sinc \] or

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+

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  • Gr (2.36)
@@ -1755,7 +1755,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-21 Mon 20:41

+

Created: 2022-03-01 Tue 08:14

diff --git a/build/ems_es_ep_d.html b/build/ems_es_ep_d.html index 0acf871..92cad2e 100644 --- a/build/ems_es_ep_d.html +++ b/build/ems_es_ep_d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,14 +1628,14 @@ field is independent of the path, and that it is proportional to the test charge We can thus define a function representing the work per unit charge and called the electrostatic potential

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  • Gr (2.21)
@@ -1651,7 +1651,7 @@ called the electrostatic potential \]

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