From 21bf9fdba59f5bfd9f9a12e8a3b106f863bdff63 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jean-S=C3=A9bastien?= Date: Wed, 2 Mar 2022 15:47:54 +0100 Subject: [PATCH] Update 2022-03-02 15:47 --- 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 | 131 +++++++++++++++------ build/emd_Fl_e.html | 96 ++++++++------- build/emd_Fl_i.html | 176 ++++++++++++++++++++-------- build/emd_Fl_ief.html | 105 +++++++++++------ build/emd_Me.html | 4 +- build/emd_Me_Me.html | 75 ++++++++---- build/emd_Me_dc.html | 69 +++++++---- build/emd_Me_ebM.html | 33 ++++-- build/emd_Me_mc.html | 8 +- build/emd_ce.html | 12 +- build/emd_ce_amom.html | 31 +++-- build/emd_ce_ce.html | 21 ++-- build/emd_ce_mom.html | 52 +++++++-- build/emd_ce_mst.html | 104 +++++++++++------ build/emd_ce_poy.html | 220 +++++++++++++++++++++++++---------- build/emd_emw.html | 12 +- build/emd_emw_ep.html | 8 +- build/emd_emw_mpw.html | 6 +- build/emd_emw_we.html | 6 +- build/emdm.html | 4 +- build/emdm_Me.html | 4 +- build/emdm_Me_Mem.html | 20 ++-- build/emdm_Me_bc.html | 6 +- build/emdm_emwm.html | 4 +- build/emdm_emwm_ad.html | 4 +- build/emdm_emwm_ad_c.html | 4 +- build/emdm_emwm_plm.html | 6 +- build/emdm_emwm_refl.html | 4 +- build/emdm_emwm_refl_Ba.html | 4 +- build/emdm_emwm_refl_Fe.html | 4 +- build/emdm_emwm_refl_ni.html | 4 +- build/emdm_emwm_refl_oi.html | 12 +- 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 | 8 +- build/emf_svp.html | 12 +- build/ems.html | 4 +- build/ems_ca.html | 12 +- build/ems_ca_fe.html | 12 +- build/ems_ca_fe_L.html | 42 +++---- 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 | 22 ++-- 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 | 6 +- 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 | 8 +- 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 | 10 +- 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_dcB.html | 4 +- build/ems_ms_lf.html | 4 +- build/ems_ms_lf_pc.html | 20 ++-- build/ems_ms_vp.html | 4 +- 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_msm.html | 12 +- build/emsm_msm_H.html | 4 +- build/emsm_msm_H_A.html | 50 ++++---- 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 | 10 +- build/red_rem_Ltf.html | 6 +- build/red_rem_Me.html | 16 +-- build/red_rem_mre.html | 6 +- build/red_rm.html | 4 +- build/red_rm_Mf.html | 10 +- build/red_rm_pt.html | 8 +- build/red_rm_rme.html | 12 +- build/red_sr.html | 4 +- build/red_sr_4v.html | 10 +- build/red_sr_Lt.html | 6 +- build/red_sr_p.html | 8 +- 194 files changed, 1653 insertions(+), 1216 deletions(-) diff --git a/build/a.html b/build/a.html index 0649223..2b8f885 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/a_l.html b/build/a_l.html index 1b20b73..b62028d 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c.html b/build/c.html index 51077b5..b707102 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m.html b/build/c_m.html index 8dfbb61..c915d1d 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index 0ec2efc..598269e 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index 627bdf0..5f918a5 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
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+

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

-
+
  • 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index 24d66c1..0a99907 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 4b25f18..2a57f3a 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:

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+

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

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

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

-
+
@@ -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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index e17bf29..2487351 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index 9be1730..b6627b8 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index 548b752..c2f8a18 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
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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 @@ -1634,44 +1634,44 @@ given by the Laplacian of the corresponding vector elements.

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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
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+
+
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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 052755a..9bd3758 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index 474b607..1898372 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index aa02c9f..1c09916 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index e81501f..10ca8ec 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:

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+

-
+
  • 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_dd.html b/build/c_m_dd.html index 4b7c36d..dcfa470 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index 7a47f7d..9a0a979 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 7856927..039f37a 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index bf2eca5..eb0dd84 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic.html b/build/c_m_ic.html index 6fb9c66..f0f2dd2 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: 2022-03-01 Tue 08:14

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            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index 0880cb2..6626b73 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: 2022-03-01 Tue 08:14

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            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index 886077c..7104a48 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: 2022-03-01 Tue 08:14

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            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index acc38b9..fcfbf19 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index 144a4da..e70b4a1 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index 3e3c9b0..77bbe1e 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index efb3a98..65f9b7b 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_uf.html b/build/c_m_uf.html index 6a2426a..300e671 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 0330234..a4e0740 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index e7bacff..10c5434 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 6ffe56f..00d774c 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va.html b/build/c_m_va.html index b7de0c2..bf4a89b 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index 2f32b27..2cee2c4 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index f51c9e7..ed2041a 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index 8c49e37..a54c6b7 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index df3f07f..93bb8c8 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index d7bd155..b73e30d 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_vf.html b/build/c_m_vf.html index b55e604..f7c97e1 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index ecf3737..60fd939 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index c027a39..98d10c2 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

            -

            Created: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d.html b/build/d.html index d2ca20b..a67c2f0 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d_emd.html b/build/d_emd.html index fbc019f..f5d8152 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index f97d647..8c8090b 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index 045efcc..e0001ce 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d_emf.html b/build/d_emf.html index 12f9c51..84bca64 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: 2022-03-01 Tue 08:14

            +

            Created: 2022-03-02 Wed 15:45

            diff --git a/build/d_ems.html b/build/d_ems.html index 4e95169..874e6f0 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: 2022-03-01 Tue 08:14

          +

          Created: 2022-03-02 Wed 15:45

          diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index 3069b1b..1bbeb0a 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: 2022-03-01 Tue 08:14

          +

          Created: 2022-03-02 Wed 15:45

          diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 47e909a..bc92577 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: 2022-03-01 Tue 08:14

          +

          Created: 2022-03-02 Wed 15:45

          diff --git a/build/d_emsm.html b/build/d_emsm.html index 3f23763..787dbe4 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: 2022-03-01 Tue 08:14

          +

          Created: 2022-03-02 Wed 15:45

          diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index e6fecba..fee4460 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/d_m.html b/build/d_m.html index 301780e..012a306 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/d_red.html b/build/d_red.html index c214200..5edb9c6 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd.html b/build/emd.html index c2f9e6f..353ea41 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Fl.html b/build/emd_Fl.html index 7b50334..7efd9a6 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index 40f2617..0c1c2eb 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,15 +1622,11 @@ Table of contents emd.Fl.Fl

        -1831: 3 experiments by Faraday (according to Griffiths! but it's historically incorrect) -\paragraph{1)} Pull a loop of wire through a magnetic field. -\paragraph{2)} Move magnet around a still loop. -\paragraph{3)} Change strength of field, holding magnet and loop still. +Around 1831, Faraday performed a number of experiments pertaining to +the effects of time-dependent fields.

        -

        -Actually, historically, things didn't happen like that. The first experiment that Faraday performed (1831) involved two metal coils wound on opposite sides of a metal ring. When a current was turned on through the first coil, it generated a transient current in the second coil (as measured by a @@ -1647,27 +1643,39 @@ on this idea. Faraday observed transient current in a circuit when:

        Faraday's big insight was to summarize these effects by noticing that -

        - -

        \[ - \boxed{ - \mbox{\bf A changing magnetic field induces an electric field} - } +\boxed{ +\mbox{A changing magnetic field induces an electric field.} +} \] -

        - -

        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), - \[ - {\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}. - \] +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.9)
          • +
        + +
        + +
        +

        +\[ +{\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}. +\tag{elmofo}\label{elmofo} +\]

        @@ -1676,21 +1684,56 @@ You can think of the emf in different ways. It's the energy accumulated as a uni

        -The precise statement is that the electromotive force is proportional +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)
          • +
        + +
        + +
        +

        \[ {\cal E} = \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} -\label{Gr(7.14)} +\tag{Fl_flux}\label{Fl_flux} \] so we obtain

        -
        +

        Faraday's law (integral form N.B.: for a stationary loop) - \[ - \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a} - \label{Gr(7.15)} - \] +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.15)
          • +
        + +
        + +
        +

        +\[ +\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a} +\tag{Fl_int}\label{Fl_int} +\]

        @@ -1702,20 +1745,36 @@ for any loop (on a wire or not). Using Stokes' theorem, \] we obtain

        -
        +
        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.16)
          • +
        + +
        + +

        Faraday's law (differential form) - \[ - {\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} - \label{Gr(7.16)} - \] +\[ +{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} +\tag{Fl}\label{Fl} +\]

        -Right-hand rule always sorts signs out. Easier rule: {\bf Lenz's law}, which -states that {\bf nature resists a change in flux}. This is in fact just -{\bf Le Ch\^atelier's principle} of any action at an equilibrium point leading +Right-hand rule always sorts signs out. Easier rule: Lenz's law, which +states that physical systems naturally resist a change in flux. +This is in fact just +Le Châtelier's principle of any action at an equilibrium point leading to an opposing counter-reaction.

        @@ -1739,7 +1798,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index 750df8a..11abc16 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1631,7 +1631,7 @@ Start from zero current, integrate in time: W = \frac{1}{2} L I^2 \label{Gr(7.29)} \] -Nicer way (generalizable to surface and volume currents): from (\ref{Gr(7.25)}), flux through loop is \(\Phi = L I\). But +Nicer way (generalizable to surface and volume currents): from PLI, flux through loop is \(\Phi = L I\). But \[ \Phi = \int_{\cal S} {\bf B} \cdot d{\bf a} = \int_{\cal S} ({\boldsymbol \nabla} \times {\bf A}) \cdot d{\bf a} = \oint_{\cal P} {\bf A} \cdot d{\bf l}, @@ -1647,16 +1647,16 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo \] Generalization to volume currents:

        -
        +

        -
        +
          -
        • gr (7.31)
          • +
        • Gr (7.31)
        @@ -1664,31 +1664,22 @@ Generalization to volume currents:

        \[ -W = \frac{1}{2} \int_{\cal V} ({\bf A} \cdot {\bf J}) d\tau +W = \frac{1}{2} \int_{\cal V} d\tau ~({\bf A} \cdot {\bf J}) \tag{W_intAJ}\label{W_intAJ} \] Even better: use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\): \[ -W = \frac{1}{2\mu_0} \int_{\cal V} {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) d\tau +W = \frac{1}{2\mu_0} \int_{\cal V} d\tau ~{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) \label{Gr(7.32)} \] Integrate by parts using product rule 6: \[ -{\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf A}) -

        -
          -
        • {\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}),
          • -
        -

        +{\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}), \] so \[ -{\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}) = {\bf B} ⋅ {\bf B} -

        -
          -
        • {\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}).
          • -
        -

        +{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) = +{\bf B} \cdot {\bf B} - {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}). \] Then, \[ @@ -1698,12 +1689,27 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold \] We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with

        -
        +
        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.34)
          • +
        + +
        + +

        \[ - W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 - \label{Gr(7.34)} - \] +W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 +\tag{W_intBsq}\label{W_intBsq} +\]

        @@ -1713,26 +1719,38 @@ We can integrate over all space: after neglecting boundary terms (assuming fiel Summary: energy in electric and magnetic fields:

        \begin{align} -W_{elec} = \frac{1}{2} \int d\tau V\rho = \frac{\varepsilon_0}{2} \int d\tau E^2, \hspace{2cm} -\mbox{(2.43 and 2.45)}, \\ -W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int d\tau B^2, -\hspace{2cm} \mbox{(7.31 and 7.34)} +W_{elec} &= \frac{1}{2} \int d\tau ~V\rho &= \frac{\varepsilon_0}{2} \int d\tau ~E^2, \\ +W_{mag} &= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &= \frac{1}{2\mu_0} \int d\tau ~B^2, \end{align} - -

        -\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\). -Find energy stored in section of length \(l\). -\paragraph{Solution:} from Ampère, +which are equations W_vcd, W_intEsq, W_intAJ and W_intBsq. +

        + +
        +

        +Example: energy in coaxial cable +

        + +

        +Consider a coaxial cable with inner cylinder radius \(a\), outer \(b\), +carrying current \(I\). +

        + +

        +Task: find the energy stored in a section of length \(l\). +

        + +

        +Solution: from Ampère, \[ - {\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a < s < b, \hspace{1cm} - {\bf B} = 0, \hspace{1cm} s < a ~\mbox{or}~ s > b. - \] +{\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a < s < b, \hspace{1cm} +{\bf B} = 0, \hspace{1cm} s < a ~\mbox{or}~ s > b. +\] Energy is thus \[ - W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2 - = \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}. - \] +W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2 += \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}. +\]

        @@ -1758,7 +1776,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index 261d578..16cf28d 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1629,7 +1629,7 @@ is (using fact that \({\bf B}_1\) is proportional to \(I_1\)) \Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 \Longrightarrow \Phi_2 = M_{21} I_1 \] -where \(M_{21}\) is the {\bf mutual inductance} of the two loops. +where \(M_{21}\) is the mutual inductance of the two loops.

        @@ -1638,7 +1638,7 @@ Useful formula: \Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 = \int ({\boldsymbol \nabla} \times {\bf A}_1) \cdot d{\bf a}_2 = \oint {\bf A}_1 \cdot d{\bf l}_2 \] -But from (\ref{Gr(5.63)}), +But from A_CoulG, \[ {\bf A}_1 ({\bf r}) = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_1} \frac{d{\bf l}_1}{|{\bf r} - {\bf r}_1|} \] @@ -1647,44 +1647,89 @@ so \Phi_2 = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_2} d{\bf l}_2 \cdot \left(\oint_{{\cal P}_1} \frac{d{\bf l}_1 }{|{\bf r}_2 - {\bf r}_1|}\right) \] -and we can write the mutual inductance as the {\bf Neumann formula}, +and we can write the mutual inductance as the Neumann formula, +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.22)
          • +
        + +
        + +
        +

        \[ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}_1 \cdot d{\bf l}_2} {|{\bf r}_1 - {\bf r}_2|} -\label{Gr(7.22)} +\tag{Neumann_M}\label{Neumann_M} \] Two things: first, \(M_{21}\) is purely geometrical. Second, +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.23)
          • +
        + +
        + +
        +

        \[ M_{12} = M_{21} -\label{Gr(7.23)} +\tag{Msym}\label{Msym} \]

        -
        +

        -\paragraph{Example 7.10:} -short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside +Example: solenoid in solenoid +

        + +

        +Consider a short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) +which lies concentrically inside a very long solenoid (radius \(b\), \(n_2\) turns per unit length). Current \(I\) in short solenoid. -What is flux through long solenoid ? -\paragraph{Solution:} complicated to calculate \({\bf B}_1\). Use mutual inductance, starting from +

        + +

        +Task: compute the flux through the long solenoid. +

        + +

        +Solution: it's complicated to calculate \({\bf B}_1\). +Use mutual inductance, starting from the reverse situation: current \(I\) on outer solenoid, calculate flux through inner one. -Field of outer solenoid: from (\ref{Gr(5.57)}), +Field of outer solenoid: from Amp_int, \[ - B = \mu_0 n_2 I - \] +B = \mu_0 n_2 I +\] so flux through a single loop of inner solenoid is \[ - B \pi a^2 = \mu_0 n_2 I \pi a^2. - \] +B \pi a^2 = \mu_0 n_2 I \pi a^2. +\] For \(n_1 l\) turns in total, total flux through inner solenoid is \[ - \Phi = \mu_0 \pi a^2 n_1 n_2 l I. - \] +\Phi = \mu_0 \pi a^2 n_1 n_2 l I. +\] Same as flux through outer solenoid if inner one has current \(I\). Mutual inductance is here \[ - M = \mu_0 \pi a^2 n_1 n_2 l. - \] +M = \mu_0 \pi a^2 n_1 n_2 l. +\]

        @@ -1697,62 +1742,101 @@ What if we vary current in loop 1? Flux in 2 will vary. Induces EMF in loop 2: \label{Gr(7.24)} \] Changing current also induces EMF in the source loop itself: +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.25)
          • +
        + +
        + +
        +

        \[ \Phi = L I -\label{Gr(7.25)} +\tag{PLI}\label{PLI} \] -where \(L\) is the {\bf self-inductance} (or inductance) of the loop. Depends only on +where \(L\) is the self-inductance (or inductance) of the loop. Depends only on geometry. Changing current induces EMF of \[ {\cal E} = -L \frac{dI}{dt} \label{Gr(7.26)} \] -Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\). +Inductance: measured in henries (\(H\)). \(H = V s/A\).

        -
        +

        -\paragraph{Example 7.11:} find self-inductance of toroidal coil with +Example: self-inductance of toroidal coil +

        + +

        +Consider a toroidal coil with rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\)) which carries total of \(N\) turns. -\paragraph{Solution:} magnetic field inside toroid is (\ref{Gr(5.58)}) +

        + +

        +Task: find its self-inductance +

        + +

        +Solution: magnetic field inside toroid is Btor \[ - B = \frac{\mu_0 NI}{2\pi s} - \] +B = \frac{\mu_0 NI}{2\pi s} +\] Flux through single turn: \[ - \int {\bf B} \cdot d{\bf a} = \frac{\mu_0 N I}{2\pi} h \int_a^b \frac{ds}{s} - = \frac{\mu_0 N I h}{2\pi} \ln \frac{b}{a}. - \] +\int {\bf B} \cdot d{\bf a} = \frac{\mu_0 N I}{2\pi} h \int_a^b \frac{ds}{s} += \frac{\mu_0 N I h}{2\pi} \ln \frac{b}{a}. +\] Total flux: \(N\) times this, so self-inductance is \[ - L = \frac{\mu_0 N^2 h}{2\pi} \ln \frac{b}{a} - \label{Gr(7.27)} - \] +L = \frac{\mu_0 N^2 h}{2\pi} \ln \frac{b}{a} +\label{Gr(7.27)} +\]

        -Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}. +Inductance (like capacitance) is intrinsically positive. Use Lenz law. +Think of back EMF.

        -
        +

        -\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). -What is the current ? -\paragraph{Solution:} +Example: circuit +

        + +

        +Consider a circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\). +

        + +

        +Task: find the current +

        + +

        +Solution: Ohm's law: \[ - {\cal E}_0 - L \frac{dI}{dt} = IR \Longrightarrow I(t) = \frac{{\cal E}_0}{R} + k e^{-(R/L)t}. - \] +{\cal E}_0 - L \frac{dI}{dt} = IR \Longrightarrow I(t) = \frac{{\cal E}_0}{R} + k e^{-(R/L)t}. +\] If initial condition: \(I(0) = 0\), then \[ - I(t) = \frac{{\cal E}_0}{R} \left[ 1 - e^{-(R/L)t} \right] - \label{Gr(7.28)} - \] -where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit. +I(t) = \frac{{\cal E}_0}{R} \left[ 1 - e^{-(R/L)t} \right] +\label{Gr(7.28)} +\] +where \(\tau \equiv L/R\) is the time constant of the circuit.

        @@ -1777,7 +1861,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index dd2af9b..5c6e2a3 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,7 +1628,7 @@ Two sources of electric fields: electric charges, and changing magnetic fields.

        Electric fields induced by a changing magnetic field are determined in an exactly parallel way as magnetostatic fields from the current: exploit parallel -between Ampère and Faraday! +between Ampère and Faraday \[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} \hspace{3cm} @@ -1645,42 +1645,60 @@ law in integral form: -

        +

        -{\bf Example 7.7:} -\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ? -\paragraph{Solution:} +Example: loop with time-dependent flux +

        + +

        +Consider a time-dependent magnetic field \({\bf B}(t)\) directed vertically +through a horizontal circular region of radius \(R\). +

        + +

        +Task: find the induced \({\bf E}(t)\). +

        + +

        +Solution: amperian loop of radius \(s\), apply Faraday: \[ - \oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt} - \Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}. - \] +\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt} +\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}. +\] 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 \(a < b\), -pointing up. Field turned off. What happens ? -\paragraph{Solution:} the wheel starts spinning to compensate the reduction of field. +Example: wheel with charged rim traversed by flux +

        + +

        +Consider a wheel of radius \(b\) with line charge \(\lambda\) on the rim. +A uniform magnetic field \({\bf B}_0\) pointing up is traversing the central region +up to radius \(a < b\). The field is then turned off. What happens? +

        + +

        +Solution: the wheel starts spinning to compensate the reduction of field. Faraday: \[ - \oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt} - \Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}. - \] +\oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt} +\Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}. +\] Torque on segment \(d{\bf l}\): \(|{\bf r} \times {\bf F}| = b \lambda E dl\). Total torque: \[ - N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt} - \] -so total angular momentum imparted is +N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt} +\] +so total angular momentum imparted to the wheel is \[ - \int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0. - \] +\int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0. +\]

        @@ -1690,33 +1708,44 @@ The precise way the field is turned off doesn't matter. Only electric field doe

        -{\bf N.B.:} we use magnetostatic formulas for changing fields. This is -called the {\bf quasistatic} approximation, and works provided we deal with -'slow enough' phenomena. +N.B.: we use magnetostatic formulas for changing fields. This is +called the 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. -\paragraph{Solution:} quasistatic: magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\) +Example: field from wire with time-dependent current +

        + +

        +Consider an infinitely long straight wire which carries current \(I(t)\). +

        + +

        +Task: find the induced \({\bf E}\) field as a function of distance \(s\) from wire. +

        + +

        +Solution: assuming we can use the quasistatic approximation, the +magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\) and circles the wire. Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel to wire. Amperian loop with sides at distances \(s_0\) and \(s\): \[ - \oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a} - = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'} - = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0). - \] +\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a} += -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'} += -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0). +\] So: \[ - {\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x} - \label{Gr(7.19)} - \] +{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x} +\label{Gr(7.19)} +\] where \(K\) is a constant (depends on the history of \(I(t)\)).

        -{\bf N.B.:} this can't be true always, since it blows up as \(s \rightarrow \infty\). +N.B.: this can't be true always, since it blows up as \(s \rightarrow \infty\). Reason: in this case, we've overstepped the quasistatic limit. We need \(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).

        @@ -1743,7 +1772,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Me.html b/build/emd_Me.html index 72450b8..743a024 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index e552283..5c73a75 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,53 +1624,79 @@ Table of contents

        Full set of equations for the electromagnetic field:

        -
        +

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

        +
        +

        + + + + +

        +
        + +
        + +
        \begin{align} - (i) {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ - (ii) {\boldsymbol \nabla} \cdot {\bf B} &= 0, \hspace{1cm} &\mbox{anonymous} \nonumber \\ - (iii) {\boldsymbol \nabla} \times {\bf E} &= -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\ - (iv) {\boldsymbol \nabla} \times {\bf B} &= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &\mbox{Ampère + Maxwell}. - \label{Gr(7.39)} +(i)~ {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ +(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &= 0, &\mbox{anonymous} \nonumber \\ +(iii)~ {\boldsymbol \nabla} \times {\bf E} &= -\frac{\partial {\bf B}}{\partial t}, &\mbox{Faraday}, \nonumber \\ +(iv)~ {\boldsymbol \nabla} \times {\bf B} &= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, &\mbox{Ampère + Maxwell}. +\tag{Max_vac}\label{Max_vac} \end{align}

        Complement:

        -
        +

        -{\bf Force law} +Force law LorFo \[ - {\bf F} = q ({\bf E} + {\bf v} \times {\bf B}). - \label{Gr(7.40)} - \] +{\bf F} = q ({\bf E} + {\bf v} \times {\bf B}). +\label{Gr(7.40)} +\]

        -These equations summarize the {\bf entire content of classical electrodynamics}. +These equations contain the entirety of pre-quantum electrodynamics.

        -\paragraph{Note:} even the continuity equation can be derived from Maxwell's equations: -take divergence of \((iv)\). +Note: even the continuity equation can be derived from Maxwell's equations: +take divergence of \((iv)\) and use \((i)\).

        -

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

        -
        +
        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.42)
          • +
        + +
        + +
        \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, \\ - (ii) &{\boldsymbol \nabla} \cdot {\bf B} = 0, - &(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = \mu_0 {\bf J}, - \label{Gr(7.42)} +(i)~ {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \nonumber \\ +(ii)~{\boldsymbol \nabla} \cdot {\bf B} &= 0, \nonumber \\ +(iii)~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &= 0, \nonumber \\ +(iv)~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &= \mu_0 {\bf J}, +\tag{Max_vac_s}\label{Max_vac_s} \end{align}
        @@ -1679,6 +1705,7 @@ Better way of writing: all fields on left, all sources on right, +

        @@ -1694,7 +1721,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index 9d6d1ff..4bd6581 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,8 +1622,7 @@ Table of contents emd.Me.dc

        -The term which should be zero (but isn't) in (\ref{Gr(7.35)}) can be rewritten using -the continuity equation as +The term which should be zero (but isn't) in divcurlB can be rewritten using the continuity equation as \[ {\boldsymbol \nabla} \cdot {\bf J} = -\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial t} (\varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}) = -{\boldsymbol \nabla} \cdot \left( @@ -1631,37 +1630,67 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

        -
        +
        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.36)
          • +
        + +
        + +

        \[ - {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} - \label{Gr(7.36)} - \] +{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} +\tag{AmpMax}\label{AmpMax} +\]

        -\paragraph{Note:} this changes nothing in magnetostatics. Aesthetic appeal: +Note: this changes nothing in magnetostatics. Aesthetic appeal: \[ - \boxed{ - \mbox{A changing electric field induces a magnetic field.} - } +\boxed{ +\mbox{A changing electric field induces a magnetic field.} +} \] Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation of electromagnetic waves.

        - -

        Maxwell baptized this term the

        -
        +
        +

        +Displacement current +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.37)
          • +
        + +
        + +

        -{\bf Displacement current} \[ - {\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}. - \label{Gr(7.37)} - \] +{\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}. +\tag{Jd}\label{Jd} +\]

        @@ -1675,7 +1704,7 @@ where \(A\) is the area. Between the plates, \[ \frac{\partial E}{\partial t} = \frac{1}{\varepsilon_0 A} \frac{dQ}{dt} = \frac{1}{\varepsilon_0 A} I. \] -Checking (\ref{Gr(7.36)}), +Checking AmpMax, \[ \oint {\bf B} \cdot d{\bf l} = \mu_0 I_{\mbox{enc}} + \mu_0 \varepsilon_0 \int d{\bf a} \cdot \frac{\partial {\bf E}}{\partial t} \label{Gr(7.38)} @@ -1703,7 +1732,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index 88f3738..2599061 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,10 +1625,10 @@ Table of contents We've encountered:

        \begin{align} -(i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ -(ii) &{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &\mbox{anonymous} \nonumber \\ -(iii) &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &\mbox{Faraday}, \nonumber \\ -(iv) &{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}, \hspace{1cm} &\mbox{Ampère}. +(i)~ {\boldsymbol \nabla} \cdot {\bf E} &= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &\mbox{Gauss}, \nonumber \\ +(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &= 0, &\mbox{anonymous} \nonumber \\ +(iii)~ {\boldsymbol \nabla} \times {\bf E} &= -\frac{\partial {\bf B}}{\partial t}, &\mbox{Faraday}, \nonumber \\ +(iv)~ {\boldsymbol \nabla} \times {\bf B} &= \mu_0 {\bf J}, &\mbox{Ampère}. \end{align}

        Fatal inconsistency: div of curl must always vanish. Check on \((iii)\): @@ -1637,11 +1637,28 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\): = {\boldsymbol \nabla} \cdot \left( -\frac{\partial {\bf B}}{\partial t} \right) = -\frac{\partial}{\partial t} ({\boldsymbol \nabla} \cdot {\bf B}) = 0. \] But: try same with \((iv)\): +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (7.35)
          • +
        + +
        + +
        +

        \[ {\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} \times {\bf B}) = \mu_0 {\boldsymbol \nabla} \cdot {\bf J} -\label{Gr(7.35)} +\tag{divcurlB}\label{divcurlB} \] -LHS must be zero, but RHS is not zero for non-steady currents. Cannot be right ! +LHS must be zero, but RHS is not zero for non-steady currents. Cannot be right!

        @@ -1673,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index 499a8a7..c62a9e2 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,7 +1633,7 @@ In free space, where \(\rho\) and \({\bf J}\) vanish:

        Symmetry: replace \({\bf E}\) by \({\bf B}\) and \({\bf B}\) by \(-\mu_0 \varepsilon_0{\bf E}\) in the first pair. They turn into the second pair. This symmetry is spoiled by \(\rho\) and \({\bf J}\). What if we had -a truly symmetric situation, {\it i.e.} +a truly symmetric situation, i.e.

        \begin{align} (i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho_e}{\varepsilon_0}, @@ -1650,7 +1650,7 @@ of magnetic charge. Both charges would be conserved: {\boldsymbol \nabla} \cdot {\bf J}_e = -\frac{\partial \rho_e}{\partial t}. \label{Gr(7.44)} \] -Maxwell's equations {\bf beg} for magnetic charges. But we've never found any! +Maxwell's equations beg for magnetic charges. But we've never found any!

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

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce.html b/build/emd_ce.html index 9075b9d..ea01746 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 04223ec..310e885 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,14 +1624,29 @@ Table of contents

        The angular momentum of EM fields is directly given by

        -
        +
        +

        +Angular momentum of EM fields +

        +
        +

        + + + + +

        +
        + +
        + +

        -{\bf Angular momentum of EM fields} \[ - {\boldsymbol l} = {\boldsymbol r} \times {\boldsymbol g} - = \varepsilon_0 {\boldsymbol r} \times - \left({\boldsymbol E} \times {\boldsymbol B}\right) - \] +{\boldsymbol l} = {\boldsymbol r} \times {\boldsymbol g} += \varepsilon_0 ~{\boldsymbol r} \times +\left({\boldsymbol E} \times {\boldsymbol B}\right) +\tag{lrxg}\label{lrxg} +\]

        @@ -1654,7 +1669,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index befd23a..399dafd 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,13 +1622,13 @@ Table of contents emd.ce.ce

        -Very important distinction: {\bf global} versus {\bf local} conservation of charge. +Very important distinction: global versus local conservation of charge.

        -Charge in a volume {\cal V}: +Charge in a volume \({\cal V}\): \[ -Q_{\cal V} (t) = \int_{\cal V} d\tau \rho ({\bf r}, t) +Q_{\cal V} (t) = \int_{\cal V} d\tau~ \rho ({\bf r}, t) \label{Gr(8.1)} \] Current \({\bf J}\) flowing out through boundary \({\cal S}\) of \({\cal V}\): conservation of charge means @@ -1643,13 +1643,12 @@ This means that \] Since this is true for any volume, we have (re)derived the

        -
        +

        -{\bf Continuity equation} +Continuity equation conteq \[ - \frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0 - \label{Gr(8.4)} - \] +\frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0 +\]

        @@ -1660,7 +1659,7 @@ Therefore, conservation of charge is a direct consequence of Maxwell's equations

        One thing to note: we have viewed \(\rho\) and \({\boldsymbol J}\) as sources (the ''right-hand side'') of Maxwell's equations. The continuity equation thus -imposes a functional constraint on these sources: not {\it any} \(\rho\) and +imposes a functional constraint on these sources: not any \(\rho\) and \({\boldsymbol J}\) will do the trick, only the ones what obey it.

        @@ -1684,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index 3995e55..28956ec 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ Table of contents

        From Newton's second law, \[ - {\boldsymbol F} = \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt} +{\boldsymbol F} = \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt} \] we have \[ @@ -1636,24 +1636,54 @@ 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 +

        +
        +

        + + + + +

        +
        + +
        + +

        -{\bf Momentum density in the EM fields} \[ - {\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B} - \] +{\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B} +\tag{gExB}\label{gExB} +\]

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

        -
        +
        +

        +Continuity equation for EM momentum +

        +
        +

        + + + + +

        +
        + +
        + +

        -{\bf Continuity equation for EM momentum} \[ - \frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0 - \] +\frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0 +\tag{contg}\label{contg} +\]

        @@ -1677,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index fcb15eb..c8cd786 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,30 +1638,27 @@ Substitute for \(\rho\) and \({\boldsymbol J}\) using Maxwell (Gauss and Ampère

        On the other hand we have -\[ - \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B} \right) - = \frac{∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}

        -
          -
        • {\boldsymbol E} × \frac{∂ {\boldsymbol B}}{∂ t}.
          • -
        +\begin{equation} + \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right) + = \frac{\partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B} + + {\boldsymbol E} \times \frac{\partial {\boldsymbol B}}{\partial t}. +\end{equation}

        -\] Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\), -\[ - \frac{ ∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B} - = \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B}\right)

        -
          -
        • {\boldsymbol E} × \left({\boldsymbol ∇} × {\boldsymbol E} \right)
          • -
        +\begin{equation} + \frac{ \partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B} + = \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B}\right) + + {\boldsymbol E} \times \left({\boldsymbol \nabla} \times {\boldsymbol E} \right) +\end{equation}

        -\] so \[ {\boldsymbol f} = \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} - {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \right) - \frac{1}{\mu_0} \left( {\boldsymbol B} \times \left( {\boldsymbol \nabla} \times {\boldsymbol B} \right) \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right). \] -Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). Moreover, by product rule 4, +Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). +Moreover, by grad_sprod, \[ \frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \] @@ -1678,18 +1675,27 @@ and similarly for \({\boldsymbol B}\). We thus get

        This expression can be greatly simplified by introducing the

        -
        +

        -{\bf Maxwell stress tensor} -\[ - T_{ij} ≡ ε_0 \left( E_i E_j - \frac{1}{2} δ_{ij} E^2\right) +Maxwell stress tensor

        -
          -
        • \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} δ_{ij} B^2 \right)
          • -
        +

        -\] + + + +

        +
        + +
        + +
        +\begin{equation} +T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right) +\tag{MaxST}\label{MaxST} +\end{equation}

        @@ -1699,26 +1705,56 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction

        -We then obtain +We then obtain the

        -
        +
        +

        +EM force per unit volume +

        +
        +

        + + + + +

        +
        + +
        + +

        -{\bf EM force per unit volume} \[ - {\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t} - \] +{\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t} +\tag{fT}\label{fT} +\]

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

        -
        +
        +

        +Total force on charges in volume +

        +
        +

        + + + + +

        +
        + +
        + +

        -{\bf Total force on charges in volume} \[ - {\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau. - \] +{\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau. +\tag{totFo}\label{totFo} +\]

        @@ -1742,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index fbfea9b..d62a2a8 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,38 +1638,45 @@ Total energy should be sum of these two. Derivation from scratch.

        Suppose that at time \(t\), we have fields \({\bf E}\) and \({\bf B}\) produced by some charge and current distributions \(\rho\) and \({\bf J}\). In an interval \(dt\), how much work is -done by EM forces ? From Lorentz force law: +done by EM forces? From Lorentz force law: \[ {\bf F} \cdot d{\bf l} = q({\bf E} + {\bf v} \times {\bf B}) \cdot {\bf v} dt = q ~{\bf E} \cdot {\bf v} dt \] 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)
          • +
        + +
        + +
        +

        \[ \frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J} -\label{Gr(8.6)} +\tag{dWdt_intEJ}\label{dWdt_intEJ} \] The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume. In terms of fields alone: use Ampère-Maxwell: \[ {\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t} \] -Using product rule 6, +Using div_xprod, \[ -{\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf E}) -

        -
          -
        • {\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}),
          • -
        -

        +{\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf E}) - {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}), \] Invoking Faraday \({\boldsymbol \nabla} \times {\bf E} = - \partial {\bf B}/\partial t\), \[ -{\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}) = - {\bf B} ⋅ \frac{∂ {\bf B}}{∂ t} -

        -
          -
        • {\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).
          • -
        -

        +{\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) = - {\bf B} \cdot \frac{\partial {\bf B}}{\partial t} - {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}). \] But obviously, \[ @@ -1679,28 +1686,35 @@ But obviously, \] so we get \[ -{\bf E} ⋅ {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right) -

        -
          -
        • \frac{1}{\mu_0} {\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).
          • -
        -

        +{\bf E} \cdot {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}). \label{Gr(8.8)} \] -Substituting this in \ref{Gr(8.6)} and using the divergence theorem, +Substituting this in dWdt_intEJ and using the divergence theorem, we obtain

        -
        +

        -{\bf Poynting's theorem} -\[ - \frac{dW}{dt} = -\frac{d}{d t} ∫_{\cal V} dτ \frac{1}{2} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right) +Poynting's theorem

        -
          -
        • \frac{1}{\mu_0} \oint_{\cal S} d{\bf a} ⋅ ({\bf E} × {\bf B})
          • -
        +

        - \label{Gr(8.9)} + + + + +

        +
        +
          +
        • Gr (8.9)
          • +
        + +
        + +
        +

        +\[ +\frac{dW}{dt} = -\frac{d}{d t} \int_{\cal V} d\tau \frac{1}{2} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} \oint_{\cal S} d{\bf a} \cdot ({\bf E} \times {\bf B}) +\tag{👉Thm}\label{👉Thm} \]

        @@ -1713,41 +1727,92 @@ 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: +Energy per unit time, per unit area carried by EM fields: given by the

        -
        +
        +

        +Poynting vector +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (8.10)
          • +
        + +
        + +

        -{\bf Poynting vector} \[ - {\bf S} \equiv \frac{1}{\mu_0} ({\bf E} \times {\bf B}) - \label{Gr(8.10)} - \] +{\bf S} \equiv \frac{1}{\mu_0} ({\bf E} \times {\bf B}) +\tag{PoyntingVec}\label{PoyntingVec} +\]

        We can thus express Poynting's theorem more compactly:

        -
        +
        +

        +Poynting's theorem (integral form) +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (8.11)
          • +
        + +
        + +

        -{\bf Poynting's theorem} \[ - \frac{dW}{dt} = - \frac{dU_{em}}{dt} - \oint_{\cal S} d{\bf a} \cdot {\bf S}. - \label{Gr(8.11)} - \] +\frac{dW}{dt} = - \frac{dU_{em}}{dt} - \oint_{\cal S} d{\bf a} \cdot {\bf S}. +\tag{PoyntingThm_int}\label{PoyntingThm_int} +\]

        where we have defined the total

        -
        +
        +

        +Energy in electromagnetic fields +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (8.5)
          • +
        + +
        + +

        -{\bf Energy in electromagnetic fields} \[ - U_{em} \equiv \frac{1}{2} \int d\tau \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \label{Gr(8.5)} - \] +U_{em} \equiv \frac{1}{2} \int d\tau \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) +\tag{Uem}\label{Uem} +\]

        @@ -1764,13 +1829,30 @@ Then, \] so we get the

        -
        +
        +

        +Poynting theorem (differential form) +

        +
        +

        + + + + +

        +
        +
          +
        • Gr (8.14)
          • +
        + +
        + +

        -{\bf Poynting theorem (differential form)} \[ - \frac{\partial}{\partial t} u_{em} + {\boldsymbol \nabla} \cdot {\bf S} = 0 - \label{Gr(8.14)} - \] +\frac{\partial}{\partial t} u_{em} + {\boldsymbol \nabla} \cdot {\bf S} = 0 +\tag{PoyntingThm}\label{PoyntingThm} +\]

        @@ -1781,26 +1863,38 @@ and has a similar for to the continuity equation -
        +
        +

        +Example: Joule heating +

        + +

        +Task: characterize the energy flow for a current-carrying wire. +

        + +

        +Solution: the energy per unit time delivered to wire the wire can +be obtained from Poynting's theorem. +

        +

        -\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 \[ - {\boldsymbol E} = \frac{V}{L} \hat{\boldsymbol x}, - \] +{\boldsymbol E} = \frac{V}{L} \hat{\boldsymbol x}, +\] where \(V\) is the potential difference between the ends ald \(L\) is the length. Magnetic field is circumferential: wire of radius \(a\), \[ - {\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi} - \] +{\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi} +\] Poynting: \[ - {\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s} - \] +{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s} +\] and points radially inwards. Energy per unit time passing surface of wire: \[ - \int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I - \] +\int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I +\] where the minus sign means energy is flowing {\it in} (the wire heats up), and the value is as expected.

        @@ -1826,7 +1920,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

        Author: Jean-Sébastien Caux

        -

        Created: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_emw.html b/build/emd_emw.html index 3d28bc1..0a1b3f8 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index 3fc6cb3..d5e1548 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index 474d7a1..64a7e25 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index f21e8ac..47983d8 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emdm.html b/build/emdm.html index 51782fb..b018726 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 7eb9d64..f0df4c5 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: 2022-03-01 Tue 08:14

        +

        Created: 2022-03-02 Wed 15:45

        diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 171e7b0..c1e1f09 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

        -
        +

        {\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:

        -
      -
      - +
      + 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: 2022-03-01 Tue 08:14

      +

      Created: 2022-03-02 Wed 15:45

      diff --git a/build/ems_ca_fe.html b/build/ems_ca_fe.html index 2ba05ad..763ba50 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

      -
      +

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

      -
      +

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

      -
      +

      @@ -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

      -
      +

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

      Author: Jean-Sébastien Caux

      -

      Created: 2022-03-01 Tue 08:14

      +

      Created: 2022-03-02 Wed 15:45

      diff --git a/build/ems_ca_fe_L.html b/build/ems_ca_fe_L.html index f7ecf48..767b63f 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

      -
      +

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

      The solution to this is

      -
      +

      -
      +
      • 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

      -
      +

      -
      +
      @@ -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),

      -
      +

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

      -
      - +
      + Physicist's proof

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

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

      Theorem:

      -
      +

      -
      +
      @@ -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.

      -
      +

      -
      +
      -
      +

      Earnshaw's theorem (physical version)

      @@ -2018,7 +2018,7 @@ Going back to Poisson's equation, we can make a few comments:

      We therefore want to ask the question: under what conditions can an electrostatic problem be fully -defined by solving Poisson's equation ? +defined by solving Poisson's equation?

      @@ -2087,7 +2087,7 @@ their maximal and minimal value on the boundary, we must have \(U = 0\) \(\foral

      -This all feels a bit amateurish and not very systematic. Can we be more precise and general? What kinds of boundary information do we really need to specify the solution uniquely ? +This all feels a bit amateurish and not very systematic. Can we be more precise and general? What kinds of boundary information do we really need to specify the solution uniquely?

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

      Author: Jean-Sébastien Caux

      -

      Created: 2022-03-01 Tue 08:14

      +

      Created: 2022-03-02 Wed 15:45

      diff --git a/build/ems_ca_fe_g.html b/build/ems_ca_fe_g.html index bd863e5..d48daad 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
      -
      +

      George Green

      -
    -
    +

    -
    +
    • FLS II (4.9)
    • Gr (2.1)
      • @@ -1646,7 +1646,7 @@ 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, @@ -1663,7 +1663,7 @@ 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 @@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/ems_es_ep.html b/build/ems_es_ep.html index 0b4b99e..a9ba2a1 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: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/ems_es_ep_PL.html b/build/ems_es_ep_PL.html index 3deda56..1d5637b 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

-
+

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

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

When the charge density vanishes, it becomes more simply

-
+

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

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

Author: Jean-Sébastien Caux

-

Created: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/ems_es_ep_bc.html b/build/ems_es_ep_bc.html index feec1a6..cbfe9c0 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:

-
+

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

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-03-01 Tue 08:14

+

Created: 2022-03-02 Wed 15:45

diff --git a/build/ems_es_ep_d.html b/build/ems_es_ep_d.html index 92cad2e..acfa02c 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

-
+

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

-