From 2d1d8d39a44a3aadd0b0e2199cbfdb4e3ab8d71f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jean-S=C3=A9bastien?= Date: Tue, 22 Mar 2022 10:53:22 +0100 Subject: [PATCH] Update 2022-03-22 10:53 --- build/a.html | 4 +-- build/a_l.html | 4 +-- build/c.html | 4 +-- build/c_m.html | 4 +-- build/c_m_cs.html | 4 +-- build/c_m_cs_cyl.html | 16 +++++----- build/c_m_cs_hyp.html | 4 +-- build/c_m_cs_sph.html | 36 ++++++++++----------- build/c_m_dc.html | 4 +-- build/c_m_dc_curl.html | 4 +-- build/c_m_dc_d2.html | 38 +++++++++++----------- build/c_m_dc_del.html | 4 +-- build/c_m_dc_div.html | 4 +-- build/c_m_dc_g.html | 4 +-- build/c_m_dc_pr.html | 28 ++++++++-------- build/c_m_dd.html | 4 +-- build/c_m_dd_1d.html | 4 +-- build/c_m_dd_3d.html | 16 +++++----- build/c_m_dd_div.html | 4 +-- build/c_m_ic.html | 4 +-- build/c_m_ic_ftc.html | 4 +-- build/c_m_ic_ftg.html | 4 +-- build/c_m_ic_gauss.html | 4 +-- build/c_m_ic_ip.html | 4 +-- build/c_m_ic_lsv.html | 22 ++++++------- build/c_m_ic_stokes.html | 8 ++--- build/c_m_uf.html | 4 +-- build/c_m_uf_cyl.html | 4 +-- build/c_m_uf_sph.html | 4 +-- build/c_m_uf_vi.html | 4 +-- build/c_m_va.html | 4 +-- build/c_m_va_cp.html | 4 +-- build/c_m_va_n.html | 4 +-- build/c_m_va_pds.html | 4 +-- build/c_m_va_sp.html | 4 +-- build/c_m_va_tp.html | 4 +-- build/c_m_vf.html | 4 +-- build/c_m_vf_helm.html | 4 +-- build/c_m_vf_pot.html | 4 +-- build/d.html | 8 ++--- build/d_emd.html | 4 +-- build/d_emd_ce.html | 4 +-- build/d_emd_emw.html | 4 +-- build/d_emf.html | 4 +-- build/d_ems.html | 4 +-- build/d_ems_ca.html | 4 +-- build/d_ems_ms.html | 4 +-- build/d_emsm.html | 4 +-- build/d_emsm_msm.html | 4 +-- build/d_m.html | 4 +-- build/d_red.html | 4 +-- build/emd.html | 12 +++---- build/emd_Fl.html | 4 +-- build/emd_Fl_Fl.html | 26 +++++++-------- build/emd_Fl_e.html | 16 +++++----- build/emd_Fl_i.html | 22 ++++++------- build/emd_Fl_ief.html | 10 +++--- build/emd_Me.html | 4 +-- build/emd_Me_Me.html | 18 +++++------ build/emd_Me_dc.html | 16 +++++----- build/emd_Me_ebM.html | 8 ++--- build/emd_Me_mc.html | 4 +-- build/emd_ce.html | 12 +++---- build/emd_ce_amom.html | 10 +++--- build/emd_ce_ce.html | 6 ++-- build/emd_ce_mom.html | 16 +++++----- build/emd_ce_mst.html | 22 ++++++------- build/emd_ce_poy.html | 40 +++++++++++------------ build/emd_emw.html | 12 +++---- build/emd_emw_ep.html | 12 +++---- build/emd_emw_mpw.html | 14 ++++---- build/emd_emw_we.html | 10 +++--- build/emdm.html | 4 +-- build/emdm_Me.html | 4 +-- build/emdm_Me_Mem.html | 40 +++++++++++------------ build/emdm_Me_bc.html | 36 ++++++++++----------- build/emdm_emwm.html | 12 +++---- build/emdm_emwm_ad.html | 4 +-- build/emdm_emwm_ad_c.html | 6 ++-- build/emdm_emwm_plm.html | 16 +++++----- build/emdm_emwm_refl.html | 4 +-- build/emdm_emwm_refl_ni.html | 8 ++--- build/emdm_emwm_refl_oi.html | 28 ++++++++-------- build/emdm_emwm_refr.html | 4 +-- build/emdm_emwm_wg.html | 4 +-- build/emdm_emwm_wg_c.html | 4 +-- build/emdm_emwm_wg_gw.html | 4 +-- build/emdm_emwm_wg_r.html | 4 +-- build/emf.html | 12 +++---- build/emf_g.html | 4 +-- build/emf_g_Cg.html | 4 +-- build/emf_g_Lg.html | 28 ++++++++-------- build/emf_svp.html | 24 +++++++------- build/ems.html | 4 +-- build/ems_ca.html | 12 +++---- build/ems_ca_fe.html | 12 +++---- build/ems_ca_fe_L.html | 38 +++++++++++----------- build/ems_ca_fe_g.html | 16 +++++----- build/ems_ca_fe_uP.html | 20 ++++++------ build/ems_ca_me.html | 4 +-- build/ems_ca_me_Ed.html | 12 +++---- build/ems_ca_me_Eq.html | 4 +-- build/ems_ca_me_a.html | 20 ++++++------ build/ems_ca_me_h.html | 24 +++++++------- build/ems_ca_me_md.html | 16 +++++----- build/ems_ca_mi.html | 8 ++--- build/ems_ca_mi_fe.html | 8 ++--- build/ems_ca_mi_isc.html | 12 +++---- build/ems_ca_mi_o.html | 4 +-- build/ems_ca_sv.html | 4 +-- build/ems_ca_sv_car.html | 28 ++++++++-------- build/ems_ca_sv_cyl.html | 4 +-- build/ems_ca_sv_sph.html | 62 ++++++++++++++++++------------------ build/ems_es.html | 12 +++---- build/ems_es_c.html | 4 +-- build/ems_es_c_cap.html | 6 ++-- build/ems_es_c_ic.html | 6 ++-- build/ems_es_c_p.html | 6 ++-- build/ems_es_c_sc.html | 12 +++---- build/ems_es_e.html | 20 ++++++------ build/ems_es_ec.html | 4 +-- build/ems_es_ec_b.html | 4 +-- build/ems_es_ec_c.html | 4 +-- build/ems_es_ec_q.html | 4 +-- build/ems_es_ec_s.html | 4 +-- build/ems_es_ef.html | 4 +-- build/ems_es_ef_Gl.html | 24 +++++++------- build/ems_es_ef_cE.html | 16 +++++----- build/ems_es_ef_ccd.html | 26 +++++++-------- build/ems_es_ef_pc.html | 18 +++++------ build/ems_es_efo.html | 4 +-- build/ems_es_efo_cl.html | 14 ++++---- build/ems_es_efo_e.html | 8 ++--- build/ems_es_efo_exp.html | 20 ++++++------ build/ems_es_efo_ps.html | 6 ++-- build/ems_es_ep.html | 4 +-- build/ems_es_ep_PL.html | 16 +++++----- build/ems_es_ep_bc.html | 12 +++---- build/ems_es_ep_d.html | 44 ++++++++++++------------- build/ems_es_ep_ex.html | 8 ++--- build/ems_es_ep_fp.html | 10 +++--- build/ems_ms.html | 12 +++---- build/ems_ms_BS.html | 28 ++++++++-------- build/ems_ms_ce.html | 10 +++--- build/ems_ms_dcB.html | 4 +-- build/ems_ms_dcB_c.html | 32 +++++++++---------- build/ems_ms_dcB_d.html | 10 +++--- build/ems_ms_dcB_iw.html | 8 ++--- build/ems_ms_lf.html | 4 +-- build/ems_ms_lf_pc.html | 14 ++++---- build/ems_ms_lf_sc.html | 30 ++++++++--------- build/ems_ms_vp.html | 4 +-- build/ems_ms_vp_A.html | 40 +++++++++++------------ build/ems_ms_vp_LC.html | 18 +++++------ build/ems_ms_vp_comp.html | 4 +-- build/ems_ms_vp_mbc.html | 22 ++++++------- build/ems_ms_vp_me.html | 32 +++++++++---------- build/emsm.html | 12 +++---- build/emsm_esm.html | 4 +-- build/emsm_esm_D.html | 30 ++++++++--------- build/emsm_esm_D_bc.html | 14 ++++---- build/emsm_esm_di.html | 6 ++-- build/emsm_esm_ld.html | 4 +-- build/emsm_esm_ld_bvp.html | 30 ++++++++--------- build/emsm_esm_ld_e.html | 10 +++--- build/emsm_esm_ld_f.html | 12 +++---- build/emsm_esm_ld_sp.html | 24 +++++++------- build/emsm_esm_mE.html | 4 +-- build/emsm_esm_mE_P.html | 4 +-- build/emsm_esm_mE_o.html | 20 ++++++------ build/emsm_esm_po.html | 28 ++++++++-------- build/emsm_esm_po_fid.html | 14 ++++---- build/emsm_esm_po_pibc.html | 4 +-- build/emsm_msm.html | 12 +++---- build/emsm_msm_H.html | 4 +-- build/emsm_msm_H_A.html | 48 ++++++++++++++-------------- build/emsm_msm_a.html | 4 +-- build/emsm_msm_fmo.html | 4 +-- build/emsm_msm_fmo_bc.html | 24 +++++++------- build/emsm_msm_fmo_fim.html | 4 +-- build/emsm_msm_fmo_pibc.html | 4 +-- build/emsm_msm_lnlm.html | 4 +-- build/emsm_msm_lnlm_fm.html | 4 +-- build/emsm_msm_lnlm_sp.html | 22 ++++++------- build/emsm_msm_m.html | 4 +-- build/emsm_msm_m_dpf.html | 10 +++--- build/emsm_msm_m_fdi.html | 16 +++++----- build/in.html | 4 +-- build/in_p.html | 4 +-- build/in_t.html | 4 +-- build/in_t_c.html | 32 +++++++++---------- build/in_t_l.html | 4 +-- build/index.html | 4 +-- build/qed.html | 4 +-- build/qed_t.html | 4 +-- build/red.html | 12 +++---- build/red_rem.html | 4 +-- build/red_rem_Fmunu.html | 22 ++++++------- build/red_rem_Ltf.html | 10 +++--- build/red_rem_Me.html | 44 ++++++++++++------------- build/red_rem_mre.html | 18 +++++------ build/red_rm.html | 4 +-- build/red_rm_Mf.html | 35 +++++++++++++------- build/red_rm_pt.html | 16 +++++----- build/red_rm_rme.html | 20 ++++++------ build/red_sr.html | 4 +-- build/red_sr_4v.html | 19 ++++++----- build/red_sr_Lt.html | 6 ++-- build/red_sr_p.html | 12 +++---- 209 files changed, 1240 insertions(+), 1226 deletions(-) diff --git a/build/a.html b/build/a.html index f6c2424..9466867 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/a_l.html b/build/a_l.html index b1f2e59..88cb9b4 100644 --- a/build/a_l.html +++ b/build/a_l.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: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c.html b/build/c.html index 7054128..c78b680 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m.html b/build/c_m.html index 7e563a4..f767f07 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index dfe3b1d..883f7f2 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index 78fd440..9e9c0e6 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1659,14 +1659,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Gradient
-
+

-
+
  • Gr4(1.79)
@@ -1687,14 +1687,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Divergence
-
+

-
+
  • Gr4(2.21)
@@ -1715,14 +1715,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \
Curl
-
+

-
+
  • Gr4(2.21)
@@ -1771,7 +1771,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index a5ad65e..5e3e1be 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 29dce12..e42448c 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,14 +1623,14 @@ which \(r\) is the distance from the chosen origin, The usual Cartesian coordinates relate to spherical coordinates according to

-
+

-
+
@@ -1655,14 +1655,14 @@ A generic vector can be expressed as where the explicit relation between spherical and Cartesian unit vectors is

-
+

-
+
@@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).

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

-
+

-
+
@@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin

Infinitesimal volume element:

-
+

-
+
@@ -1736,14 +1736,14 @@ Infinitesimal surface element: depends on situation.
Gradient
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+

-
+
@@ -1760,14 +1760,14 @@ Infinitesimal surface element: depends on situation.
Divergence
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+

-
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@@ -1784,14 +1784,14 @@ Infinitesimal surface element: depends on situation.
Curl
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+

-
+
@@ -1810,14 +1810,14 @@ Infinitesimal surface element: depends on situation.
Laplacian
-
+

-
+
@@ -1851,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index 83ba3fe..17a790f 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index 394bf5b..afc4ed5 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index 248c1a9..08d4287 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1615,9 +1615,9 @@ Table of contents
-
-
Divergence of gradient
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+
+
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 @@ -1626,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.

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

This always vanishes.

-
-
Gradient of the divergence
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+
+
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
+
+

-
+
@@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 3bfe77d..12b1ffa 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index effcd20..33e2cf7 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.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: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 069c74a..6816a5c 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.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: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 50a52c1..bdf7e3a 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,14 +1622,14 @@ explicited as follows:

Gradient of a product:

-
+

-
+
  • Gr (3)
  • W (1-111)
    • @@ -1649,14 +1649,14 @@ explicited as follows:

    Gradient of a scalar product:

    -
    +

    -
    +
    • Gr (4)
    • W (1-112)
      • @@ -1676,14 +1676,14 @@ explicited as follows:

      Divergence of a product:

      -
      +

      -
      +
      • Gr (5)
      • W (1-115)
        • @@ -1703,14 +1703,14 @@ explicited as follows:

        Divergence of a cross product:

        -
        +

        -
        +
        • Gr (6)
        • W (1-116)
          • @@ -1730,14 +1730,14 @@ explicited as follows:

          Curl of a product:

          -
          +

          -
          +
          • Gr (7)
          • W (1-118)
            • @@ -1757,14 +1757,14 @@ explicited as follows:

            Curl of a cross product:

            -
            +

            -
            +
            • Gr (8)
            • W (1-119)
              • @@ -1805,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_dd.html b/build/c_m_dd.html index 4bbeedd..3d54084 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index 4d5e0a5..796ce85 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 8c3689d..6a365a5 100644 --- a/build/c_m_dd_3d.html +++ b/build/c_m_dd_3d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox: More generally,

            -
            +

            -
            +
            • Gr (1.100)
            @@ -1662,14 +1662,14 @@ More generally, Since

            -
            +

            -
            +
            • Gr (1.101)
            @@ -1685,14 +1685,14 @@ Since

            we have that

            -
            +

            -
            +
            • Gr (1.102)
            @@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index 6c47520..bb36c11 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic.html b/build/c_m_ic.html index ff7cae8..971b004 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index be85f35..760adf0 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index d916e1e..337bbdd 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index d2e0f0e..0cf78ec 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index 60d0902..9e4098d 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 8aa7651..f4d4f3a 100644 --- a/build/c_m_ic_lsv.html +++ b/build/c_m_ic_lsv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1615,9 +1615,9 @@ Table of contents
            -
            -
            Line Integrals
            -
            +
            +
            Line Integrals
            +

            \[ {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l} @@ -1646,9 +1646,9 @@ Integral over a closed loop:

            -
            -
            Surface Integrals
            -
            +
            +
            Surface Integrals
            +

            \[ \int_{\cal S} {\bf v} \cdot d{\bf a} @@ -1668,9 +1668,9 @@ Over a closed surface:

            -
            -
            Volume Integrals
            -
            +
            +
            Volume Integrals
            +

            \[ \int_{\cal V} T d\tau @@ -1711,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index f2ea5f6..cce0089 100644 --- a/build/c_m_ic_stokes.html +++ b/build/c_m_ic_stokes.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1613,14 +1613,14 @@ Table of contents c.m.ic.stokes
            -
            +

            -
            +
            • Gr (1.57)
            @@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

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            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 5e90540..8287326 100644 --- a/build/c_m_uf_cyl.html +++ b/build/c_m_uf_cyl.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index bdc5579..edab540 100644 --- a/build/c_m_uf_sph.html +++ b/build/c_m_uf_sph.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 804fe09..e82e382 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1655,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va.html b/build/c_m_va.html index beee8d7..ec806b7 100644 --- a/build/c_m_va.html +++ b/build/c_m_va.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index b1ce0df..c26ce79 100644 --- a/build/c_m_va_cp.html +++ b/build/c_m_va_cp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1670,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index 3702be0..e89a373 100644 --- a/build/c_m_va_n.html +++ b/build/c_m_va_n.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index 0d287a0..0f5c894 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index 86f2e62..5da8c41 100644 --- a/build/c_m_va_sp.html +++ b/build/c_m_va_sp.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index b466539..fe0319e 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_vf.html b/build/c_m_vf.html index b45e51e..7095019 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index c15381d..ada5f71 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index fff0045..bc1a15d 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d.html b/build/d.html index af1c780..838f5b8 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1614,8 +1614,8 @@ Table of contents d
            -
            - +
            + Objectives @@ -1679,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

            Author: Jean-Sébastien Caux

            -

            Created: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d_emd.html b/build/d_emd.html index 9f5d5fa..ea4ba30 100644 --- a/build/d_emd.html +++ b/build/d_emd.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index a8741ee..6a59509 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index d9dd3e6..cf60733 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d_emf.html b/build/d_emf.html index 496bf39..e0b9c5e 100644 --- a/build/d_emf.html +++ b/build/d_emf.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: 2022-03-15 Tue 08:10

            +

            Created: 2022-03-22 Tue 10:52

            diff --git a/build/d_ems.html b/build/d_ems.html index 9f4ce54..9647680 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-03-15 Tue 08:10

          +

          Created: 2022-03-22 Tue 10:52

          diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index fffacbe..c2368de 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.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: 2022-03-15 Tue 08:10

          +

          Created: 2022-03-22 Tue 10:52

          diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 04db7e4..f048017 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1656,7 +1656,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

          Author: Jean-Sébastien Caux

          -

          Created: 2022-03-15 Tue 08:10

          +

          Created: 2022-03-22 Tue 10:52

          diff --git a/build/d_emsm.html b/build/d_emsm.html index f13314a..fbab99f 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.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: 2022-03-15 Tue 08:10

          +

          Created: 2022-03-22 Tue 10:52

          diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index ad6aa8e..152ee22 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.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: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/d_m.html b/build/d_m.html index 0ffa14e..5fb3403 100644 --- a/build/d_m.html +++ b/build/d_m.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: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/d_red.html b/build/d_red.html index f640f0e..220efe9 100644 --- a/build/d_red.html +++ b/build/d_red.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

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd.html b/build/emd.html index 59ac15c..6c9fcbf 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1614,8 +1614,8 @@ Table of contents emd
        -
        - +
        + Prerequisites
          @@ -1624,8 +1624,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 21a9ed1..86a8ff6 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.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: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index 2eb510b..4c76482 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1644,18 +1644,18 @@ 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),

        -
        +

        -
        +
        • Gr (7.9)
        @@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations is that the electromotive force is proportional to the rate of change of the magnetic flux,

        -
        +

        -
        +
        • Gr (7.14)
        @@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux, \] so we obtain

        -
        +

        Faraday's law (integral form N.B.: for a stationary loop)

        -
        +

        -
        +
        • Gr (7.15)
        @@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

        -
        -
        +
        +

        -
        +
        • Gr (7.16)
        @@ -1790,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index 47d03b3..e02406a 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo \] Generalization to volume currents:

        -
        +

        -
        +
        • Gr (7.31)
        @@ -1681,15 +1681,15 @@ 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

        -
        -
        +
        +

        -
        +
        • Gr (7.34)
        @@ -1718,7 +1718,7 @@ W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2 which are equations W_vcd, W_intEsq, W_intAJ and W_intBsq.

        -
        +

        Example: energy in coaxial cable

        @@ -1768,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 388ff54..3c1850b 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,14 +1641,14 @@ so \] and we can write the mutual inductance as the Neumann formula,

        -
        +

        -
        +
        • Gr (7.22)
        @@ -1665,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l} Two things: first, \(M_{21}\) is purely geometrical. Second,

        -
        +

        -
        +
        • Gr (7.23)
        @@ -1687,7 +1687,7 @@ M_{12} = M_{21} \]

        -
        +

        Example: solenoid in solenoid

        @@ -1735,14 +1735,14 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2: \] Changing current also induces EMF in the source loop itself:

        -
        +

        -
        +
        • Gr (7.25)
        @@ -1765,7 +1765,7 @@ Inductance: measured in henries (\(H\)). \(H = V s/A\).

        -
        +

        Example: self-inductance of toroidal coil

        @@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of back EMF.

        -
        +

        Example: circuit

        @@ -1853,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index 3de193e..a17e094 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ law in integral form: -
        +

        Example: loop with time-dependent flux

        @@ -1664,7 +1664,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
        -
        +

        Example: wheel with charged rim traversed by flux

        @@ -1705,7 +1705,7 @@ called the quasistatic approximation, and works provided we deal with slow enough phenomena.

        -
        +

        Example: field from wire with time-dependent current

        @@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Me.html b/build/emd_Me.html index 14162d3..6580b87 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.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: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index 447b9ef..1944a1a 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,18 +1616,18 @@ Table of contents

        Full set of equations for the electromagnetic field:

        -
        +

        Maxwell's equations (in vacuum)

        -
        +

        -
        +
        @@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:

        Complement:

        -
        +

        Force law LorFo \[ @@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).

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

        -
        -
        +
        +

        -
        +
        • Gr (7.42)
        @@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index 4d54e09..3b7c053 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,15 +1622,15 @@ The term which should be zero (but isn't) in -
        +
        +

        -
        +
        • Gr (7.36)
        @@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation

        Maxwell baptized this term the

        -
        +

        Displacement current

        -
        +

        -
        +
        • Gr (7.37)
        @@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index 1c8d345..b3cbec5 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1630,14 +1630,14 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\): \] But: try same with \((iv)\):

        -
        +

        -
        +
        • Gr (7.35)
        @@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index be16b63..08849f6 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.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

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce.html b/build/emd_ce.html index b9a793e..3c77cc0 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1614,8 +1614,8 @@ Table of contents emd.ce
        -
        - +
        + Prerequisites
          @@ -1623,8 +1623,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 950723c..16b550a 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1616,18 +1616,18 @@ Table of contents

        The angular momentum of EM fields is directly given by

        -
        +

        Angular momentum of EM fields

        -
        +

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

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index f559746..d5ec10d 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1635,7 +1635,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

        -
        +

        Continuity equation conteq \[ @@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index aa90916..1510ed0 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,18 +1628,18 @@ 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

        -
        +

        Momentum density in the EM fields

        -
        +

        -
        +
        @@ -1655,18 +1655,18 @@ 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

        -
        +

        Continuity equation for EM momentum

        -
        +

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

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index 90f337a..67ffc16 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get

        This expression can be greatly simplified by introducing the

        -
        +

        Maxwell stress tensor

        -
        +

        -
        +
        @@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction

        We then obtain the

        -
        +

        EM force per unit volume

        -
        +

        -
        +
        @@ -1726,18 +1726,18 @@ We then obtain the

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

        -
        +

        Total force on charges in volume

        -
        +

        -
        +
        @@ -1770,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index cada692..75edee8 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,14 +1636,14 @@ done by EM forces? From Lorentz force law: Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,

        -
        +

        -
        +
        • Gr (8.6)
        @@ -1683,18 +1683,18 @@ so we get Substituting this in dWdt_intEJ and using the divergence theorem, we obtain

        -
        +

        Poynting's theorem

        -
        +

        -
        +
        • Gr (8.9)
        @@ -1720,18 +1720,18 @@ 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: given by the

        -
        +

        Poynting vector

        -
        +

        -
        +
        • Gr (8.10)
        @@ -1750,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the

        We can thus express Poynting's theorem more compactly:

        -
        +

        Poynting's theorem (integral form)

        -
        +

        -
        +
        • Gr (8.11)
        @@ -1780,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:

        where we have defined the total

        -
        +

        Energy in electromagnetic fields

        -
        +

        -
        +
        • Gr (8.5)
        @@ -1820,18 +1820,18 @@ Then, \] so we get the

        -
        +

        Poynting theorem (differential form)

        -
        +

        -
        +
        • Gr (8.14)
        @@ -1854,7 +1854,7 @@ and has a similar for to the continuity equation -
        +

        Example: Joule heating

        @@ -1911,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_emw.html b/build/emd_emw.html index e88cc8b..494963b 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1614,8 +1614,8 @@ Table of contents emd.emw
        -
        - +
        + Prerequisites
          @@ -1624,8 +1624,8 @@ Prerequisites
        -
        - +
        + Objectives
          @@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index 7f90b97..441d2bb 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave, \] or more succinctly:

        -
        +

        Poynting vector of a monochromatic EM wave

        -
        +

        -
        +
        • Gr (9.57)
        @@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w

        Similary, we get the

        -
        +

        Momentum density of a monochromatic EM wave \[ @@ -1729,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index 6b5f01b..665b048 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,14 +1636,14 @@ From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia \] so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and

        -
        +

        -
        +
        • Gr (9.47)
        @@ -1662,18 +1662,18 @@ 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

        -
        +

        E and B fields for a monochromatic EM plane wave

        -
        +

        -
        +
        • Gr (9.49)
        @@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index 34cd8e8..f8b6db0 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,18 +1641,18 @@ 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

        -
        +

        Wave equations for electric and magnetic fields in vacuum

        -
        +

        -
        +
        • Gr (9.41)
        @@ -1714,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emdm.html b/build/emdm.html index d8eab27..770533d 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 60938aa..95191e9 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-15 Tue 08:10

        +

        Created: 2022-03-22 Tue 10:52

        diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 42906a3..f1e2b53 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1648,18 +1648,18 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

        -
        +

        Polarization current density

        -
        +

        -
        +
        • Gr (7.48)
        @@ -1683,7 +1683,7 @@ 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:

        -
      -
      - +
      + Prerequisites
        @@ -1627,8 +1627,8 @@ Prerequisites
      -
      - +
      + Objectives
        @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

      -

      Created: 2022-03-15 Tue 08:10

      +

      Created: 2022-03-22 Tue 10:52

      diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html index d38b3c6..b6c14d5 100644 --- a/build/ems_ca_fe.html +++ b/build/ems_ca_fe.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ 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

      -
      +

      @@ -1638,7 +1638,7 @@ from calculating either the field according to p_vcd

      -
      +

      @@ -1658,7 +1658,7 @@ condition curlE0 can be expressed as th 🐟

      -
      +

      @@ -1674,7 +1674,7 @@ 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

      -
      +

      @@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

      -

      Created: 2022-03-15 Tue 08:10

      +

      Created: 2022-03-22 Tue 10:52

      diff --git a/build/ems_ca_fe_L.html b/build/ems_ca_fe_L.html index 7c40eaf..1ad2b11 100644 --- a/build/ems_ca_fe_L.html +++ b/build/ems_ca_fe_L.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable function \(\phi (x)\) and the Laplace equation reads

      -
      +

      -
      +
      @@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads

      The solution to this is

      -
      +

      -
      +
      • Gr (3.6)
      @@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function of two variables (here: \(x\) and \(y\)), so Laplace's equation now reads

      -
      +

      -
      +
      @@ -1778,14 +1778,14 @@ 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),

      -
      +

      -
      +
      @@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere \]

      -
      - +
      + Physicist's proof

      @@ -1860,8 +1860,8 @@ proving the theorem.

      -
      - +
      + Formal proof @@ -1911,14 +1911,14 @@ we get the following general

      Theorem:

      -
      +

      -
      +
      @@ -1971,19 +1971,19 @@ 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.

      -
      +

      -
      +
      -
      +

      Earnshaw's theorem (physical version)

      @@ -2102,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

      Author: Jean-Sébastien Caux

      -

      Created: 2022-03-15 Tue 08:10

      +

      Created: 2022-03-22 Tue 10:52

      diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html index 6b9c3a4..d8e622c 100644 --- a/build/ems_ca_fe_g.html +++ b/build/ems_ca_fe_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1613,11 +1613,11 @@ Table of contents ems.ca.fe.g
      -
      +

      George Green

      -
    -
    +

    -
    +
    • FLS II (4.9)
    • Gr (2.1)
      • @@ -1638,7 +1638,7 @@ 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, @@ -1655,7 +1655,7 @@ on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's la

    -
-
+

Electrical forces obey the principle of superposition: the electrostatic interaction between two charged particles is left entirely @@ -1655,7 +1655,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/ems_es_ep.html b/build/ems_es_ep.html index 1dae905..1fd28de 100644 --- a/build/ems_es_ep.html +++ b/build/ems_es_ep.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: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/ems_es_ep_PL.html b/build/ems_es_ep_PL.html index 52d227d..e3d1c13 100644 --- a/build/ems_es_ep_PL.html +++ b/build/ems_es_ep_PL.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,14 +1622,14 @@ Our two fundamental equations for the electrostatic field are \] For the electrostatic potential, by Emgp, Gauss's law becomes

-
+

-
+
  • Gr (2.24)
  • PM (2.72)
    • @@ -1638,7 +1638,7 @@ For the electrostatic potential, by Emgp,
-
+

Poisson's equation \[ @@ -1651,14 +1651,14 @@ For the electrostatic potential, by Emgp,

When the charge density vanishes, it becomes more simply

-
+

-
+
  • Gr (2.25)
  • PM (2.75)
    • @@ -1667,7 +1667,7 @@ When the charge density vanishes, it becomes more simply
-
+

Laplace's Equation \[ @@ -1703,7 +1703,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/ems_es_ep_bc.html b/build/ems_es_ep_bc.html index 174a986..2e0a3b3 100644 --- a/build/ems_es_ep_bc.html +++ b/build/ems_es_ep_bc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1650,14 +1650,14 @@ applied to a small loop straddling the surface:

We can unify both boundary conditions into a single equation:

-
+

-
+
  • Gr (2.33)
@@ -1692,14 +1692,14 @@ The gradient however inherits the discontinuity of the electrostatic field, sinc \] or

-
+

-
+
  • Gr (2.36)
@@ -1747,7 +1747,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-15 Tue 08:10

+

Created: 2022-03-22 Tue 10:52

diff --git a/build/ems_es_ep_d.html b/build/ems_es_ep_d.html index 89e5c90..f8ad046 100644 --- a/build/ems_es_ep_d.html +++ b/build/ems_es_ep_d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,14 +1620,14 @@ 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

-
+

-
+
  • Gr (2.21)
@@ -1643,7 +1643,7 @@ called the electrostatic potential \]

-