diff --git a/build/a.html b/build/a.html index 8d599ee..621c5b5 100644 --- a/build/a.html +++ b/build/a.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/a_l.html b/build/a_l.html index 4cc6c61..4894259 100644 --- a/build/a_l.html +++ b/build/a_l.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1702,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c.html b/build/c.html index bf3331e..78f9a0e 100644 --- a/build/c.html +++ b/build/c.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m.html b/build/c_m.html index 2344296..c5fa77d 100644 --- a/build/c_m.html +++ b/build/c_m.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_cs.html b/build/c_m_cs.html index 22e151a..3f874a6 100644 --- a/build/c_m_cs.html +++ b/build/c_m_cs.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_cs_cyl.html b/build/c_m_cs_cyl.html index b699323..ca99cbc 100644 --- a/build/c_m_cs_cyl.html +++ b/build/c_m_cs_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,14 +1643,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
Gradient
-
+

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

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

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_cs_hyp.html b/build/c_m_cs_hyp.html index 64e883a..eff6a67 100644 --- a/build/c_m_cs_hyp.html +++ b/build/c_m_cs_hyp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1614,7 +1614,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_cs_sph.html b/build/c_m_cs_sph.html index 1bfac1b..f17b837 100644 --- a/build/c_m_cs_sph.html +++ b/build/c_m_cs_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1727,7 +1727,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc.html b/build/c_m_dc.html index 2f3a04c..8110bf0 100644 --- a/build/c_m_dc.html +++ b/build/c_m_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_curl.html b/build/c_m_dc_curl.html index dda8fdd..b769578 100644 --- a/build/c_m_dc_curl.html +++ b/build/c_m_dc_curl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_d2.html b/build/c_m_dc_d2.html index 0df3b18..b405947 100644 --- a/build/c_m_dc_d2.html +++ b/build/c_m_dc_d2.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1599,9 +1599,9 @@ Table of contents
-
-
Divergence of gradient
-
+
+
Divergence of gradient
+

\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the Laplacian of the scalar field \(T\). The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components @@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.

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

This always vanishes.

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

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

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

This always vanishes.

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

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_del.html b/build/c_m_dc_del.html index 31ce4e4..601c287 100644 --- a/build/c_m_dc_del.html +++ b/build/c_m_dc_del.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_div.html b/build/c_m_dc_div.html index e18760d..ad4c6b8 100644 --- a/build/c_m_dc_div.html +++ b/build/c_m_dc_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_g.html b/build/c_m_dc_g.html index 131417f..42bfd3b 100644 --- a/build/c_m_dc_g.html +++ b/build/c_m_dc_g.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dc_pr.html b/build/c_m_dc_pr.html index 449a828..7b44fca 100644 --- a/build/c_m_dc_pr.html +++ b/build/c_m_dc_pr.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dd.html b/build/c_m_dd.html index d6dff68..9539924 100644 --- a/build/c_m_dd.html +++ b/build/c_m_dd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dd_1d.html b/build/c_m_dd_1d.html index a258f42..75c7d73 100644 --- a/build/c_m_dd_1d.html +++ b/build/c_m_dd_1d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dd_3d.html b/build/c_m_dd_3d.html index 821670e..72749d4 100644 --- a/build/c_m_dd_3d.html +++ b/build/c_m_dd_3d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox: More generally,

-
+

-
+
  • Gr (1.100)
@@ -1646,14 +1646,14 @@ More generally, Since

-
+

-
+
  • Gr (1.101)
@@ -1691,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_dd_div.html b/build/c_m_dd_div.html index 65bf3cf..086b891 100644 --- a/build/c_m_dd_div.html +++ b/build/c_m_dd_div.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic.html b/build/c_m_ic.html index bad6498..6765a9f 100644 --- a/build/c_m_ic.html +++ b/build/c_m_ic.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1624,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_ftc.html b/build/c_m_ic_ftc.html index 430bac4..348de4b 100644 --- a/build/c_m_ic_ftc.html +++ b/build/c_m_ic_ftc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_ftg.html b/build/c_m_ic_ftg.html index 617c800..f94a53e 100644 --- a/build/c_m_ic_ftg.html +++ b/build/c_m_ic_ftg.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1631,7 +1631,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_gauss.html b/build/c_m_ic_gauss.html index 43463c4..398e609 100644 --- a/build/c_m_ic_gauss.html +++ b/build/c_m_ic_gauss.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1625,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_ip.html b/build/c_m_ic_ip.html index b85e2a1..d335992 100644 --- a/build/c_m_ic_ip.html +++ b/build/c_m_ic_ip.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_lsv.html b/build/c_m_ic_lsv.html index da8ccb1..6a82b7f 100644 --- a/build/c_m_ic_lsv.html +++ b/build/c_m_ic_lsv.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1599,9 +1599,9 @@ Table of contents
-
-
Line Integrals
-
+
+
Line Integrals
+

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

-
-
Surface Integrals
-
+
+
Surface Integrals
+

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

-
-
Volume Integrals
-
+
+
Volume Integrals
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_ic_stokes.html b/build/c_m_ic_stokes.html index 757ac4e..636a606 100644 --- a/build/c_m_ic_stokes.html +++ b/build/c_m_ic_stokes.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1597,14 +1597,14 @@ Table of contents c.m.ic.stokes
-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_uf.html b/build/c_m_uf.html index e6bd2bb..98dfc70 100644 --- a/build/c_m_uf.html +++ b/build/c_m_uf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_uf_cyl.html b/build/c_m_uf_cyl.html index 0229e23..9ae6d3d 100644 --- a/build/c_m_uf_cyl.html +++ b/build/c_m_uf_cyl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_uf_sph.html b/build/c_m_uf_sph.html index 1b56d2f..ae2626d 100644 --- a/build/c_m_uf_sph.html +++ b/build/c_m_uf_sph.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1634,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_uf_vi.html b/build/c_m_uf_vi.html index 88e40db..427c2af 100644 --- a/build/c_m_uf_vi.html +++ b/build/c_m_uf_vi.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va.html b/build/c_m_va.html index ebb6d98..dcadc72 100644 --- a/build/c_m_va.html +++ b/build/c_m_va.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1623,7 +1623,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va_cp.html b/build/c_m_va_cp.html index f0d88a3..19dd05c 100644 --- a/build/c_m_va_cp.html +++ b/build/c_m_va_cp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va_n.html b/build/c_m_va_n.html index 884572a..fd411dc 100644 --- a/build/c_m_va_n.html +++ b/build/c_m_va_n.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va_pds.html b/build/c_m_va_pds.html index 1ffdad7..b3018c3 100644 --- a/build/c_m_va_pds.html +++ b/build/c_m_va_pds.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1681,7 +1681,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va_sp.html b/build/c_m_va_sp.html index b136fa2..10de89b 100644 --- a/build/c_m_va_sp.html +++ b/build/c_m_va_sp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_va_tp.html b/build/c_m_va_tp.html index 40b8bf8..71c53b8 100644 --- a/build/c_m_va_tp.html +++ b/build/c_m_va_tp.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_vf.html b/build/c_m_vf.html index af3c8fb..0bcd5d3 100644 --- a/build/c_m_vf.html +++ b/build/c_m_vf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_vf_helm.html b/build/c_m_vf_helm.html index a510b43..c5cb70c 100644 --- a/build/c_m_vf_helm.html +++ b/build/c_m_vf_helm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/c_m_vf_pot.html b/build/c_m_vf_pot.html index 7cbcfa2..ff702df 100644 --- a/build/c_m_vf_pot.html +++ b/build/c_m_vf_pot.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d.html b/build/d.html index c535834..2052fab 100644 --- a/build/d.html +++ b/build/d.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents d
-
- +
+ Objectives @@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emd.html b/build/d_emd.html index c2c1f56..8f9ebad 100644 --- a/build/d_emd.html +++ b/build/d_emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emd_ce.html b/build/d_emd_ce.html index 67fab85..ea8e504 100644 --- a/build/d_emd_ce.html +++ b/build/d_emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emd_emw.html b/build/d_emd_emw.html index 0aeabd4..a07c838 100644 --- a/build/d_emd_emw.html +++ b/build/d_emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emf.html b/build/d_emf.html index c13c3e9..90c8676 100644 --- a/build/d_emf.html +++ b/build/d_emf.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1636,7 +1636,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_ems.html b/build/d_ems.html index 4b9babe..0cebd8f 100644 --- a/build/d_ems.html +++ b/build/d_ems.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1658,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_ems_ca.html b/build/d_ems_ca.html index 6e6327c..82e0e8c 100644 --- a/build/d_ems_ca.html +++ b/build/d_ems_ca.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_ems_ms.html b/build/d_ems_ms.html index 5b74bf8..4670f93 100644 --- a/build/d_ems_ms.html +++ b/build/d_ems_ms.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emsm.html b/build/d_emsm.html index e7a1a6b..0185289 100644 --- a/build/d_emsm.html +++ b/build/d_emsm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1641,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_emsm_msm.html b/build/d_emsm_msm.html index 6d4822c..efc93f1 100644 --- a/build/d_emsm_msm.html +++ b/build/d_emsm_msm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/d_m.html b/build/d_m.html index 636c2a2..5167f4c 100644 --- a/build/d_m.html +++ b/build/d_m.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

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/d_red.html b/build/d_red.html index fda9a1a..b80a386 100644 --- a/build/d_red.html +++ b/build/d_red.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd.html b/build/emd.html index a904065..eef2d71 100644 --- a/build/emd.html +++ b/build/emd.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd
-
- +
+ Prerequisites
    @@ -1608,8 +1608,8 @@ Prerequisites
-
- +
+ Objectives
    @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Fl.html b/build/emd_Fl.html index f9c00dc..1403300 100644 --- a/build/emd_Fl.html +++ b/build/emd_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Fl_Fl.html b/build/emd_Fl_Fl.html index d5a26c8..c386afc 100644 --- a/build/emd_Fl_Fl.html +++ b/build/emd_Fl_Fl.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1638,7 +1638,7 @@ Empirically: the changing magnetic field induces an electric current around the circuit. This current is really driven by an electric field having a component along the wire. The line integral of this field is called the

-
+

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

-
+

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

-
+

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

Author: Jean-Sébastien Caux

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Fl_e.html b/build/emd_Fl_e.html index 9fc8f5f..dbb1b2f 100644 --- a/build/emd_Fl_e.html +++ b/build/emd_Fl_e.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold \] We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with

-
+

\[ W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2 @@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int \hspace{2cm} \mbox{(7.31 and 7.34)} \end{align} -

+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Fl_i.html b/build/emd_Fl_i.html index c8cb1e0..33b2198 100644 --- a/build/emd_Fl_i.html +++ b/build/emd_Fl_i.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1637,7 +1637,7 @@ M_{12} = M_{21} \]

-
+

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

-
+

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

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Fl_ief.html b/build/emd_Fl_ief.html index 2b8fc4d..4ca544b 100644 --- a/build/emd_Fl_ief.html +++ b/build/emd_Fl_ief.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1621,7 +1621,7 @@ law in integral form: -
+

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

-
+

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

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Me.html b/build/emd_Me.html index 7ead01c..8a96e47 100644 --- a/build/emd_Me.html +++ b/build/emd_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1622,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Me_Me.html b/build/emd_Me_Me.html index 778cf94..2b5aea1 100644 --- a/build/emd_Me_Me.html +++ b/build/emd_Me_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,7 +1600,7 @@ Table of contents

Full set of equations for the electromagnetic field:

-
+

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

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

Complement:

-
+

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

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

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Me_dc.html b/build/emd_Me_dc.html index e4246e1..70ddd53 100644 --- a/build/emd_Me_dc.html +++ b/build/emd_Me_dc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1607,7 +1607,7 @@ the continuity equation as \] The extra term would thus be eliminated if we were to put

-
+

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

Maxwell baptized this term the

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Me_ebM.html b/build/emd_Me_ebM.html index d1fbbea..e7fc2a1 100644 --- a/build/emd_Me_ebM.html +++ b/build/emd_Me_ebM.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_Me_mc.html b/build/emd_Me_mc.html index a1e6f70..dabc031 100644 --- a/build/emd_Me_mc.html +++ b/build/emd_Me_mc.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1645,7 +1645,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce.html b/build/emd_ce.html index f96d99b..ca82e79 100644 --- a/build/emd_ce.html +++ b/build/emd_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd.ce
-
- +
+ Prerequisites
    @@ -1607,8 +1607,8 @@ Prerequisites
-
- +
+ Objectives
    @@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce_amom.html b/build/emd_ce_amom.html index 2fe0b19..22c553e 100644 --- a/build/emd_ce_amom.html +++ b/build/emd_ce_amom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1600,7 +1600,7 @@ Table of contents

The angular momentum of EM fields is directly given by

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce_ce.html b/build/emd_ce_ce.html index 1cf6377..6965d77 100644 --- a/build/emd_ce_ce.html +++ b/build/emd_ce_ce.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1619,7 +1619,7 @@ This means that \] Since this is true for any volume, we have (re)derived the

-
+

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

Author: Jean-Sébastien Caux

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce_mom.html b/build/emd_ce_mom.html index 713e3ed..023a93b 100644 --- a/build/emd_ce_mom.html +++ b/build/emd_ce_mom.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM

This is thus simply a conservation law for momentum, with

-
+

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

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

-
+

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

Author: Jean-Sébastien Caux

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Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce_mst.html b/build/emd_ce_mst.html index 5ee898d..da47ab1 100644 --- a/build/emd_ce_mst.html +++ b/build/emd_ce_mst.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get

This expression can be greatly simplified by introducing the

-
+

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

We then obtain

-
+

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

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

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_ce_poy.html b/build/emd_ce_poy.html index 855f3ce..41c8b1d 100644 --- a/build/emd_ce_poy.html +++ b/build/emd_ce_poy.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1666,7 +1666,7 @@ so we get Substituting this in \ref{Gr(8.6)} and using the divergence theorem, we obtain

-
+

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

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

-
+

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

We can thus express Poynting's theorem more compactly:

-
+

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

where we have defined the total

-
+

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

-
+

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

+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_emw.html b/build/emd_emw.html index c4a744d..977478c 100644 --- a/build/emd_emw.html +++ b/build/emd_emw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1598,8 +1598,8 @@ Table of contents emd.emw
-
- +
+ Prerequisites
    @@ -1608,8 +1608,8 @@ Prerequisites
-
- +
+ Objectives
    @@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_emw_ep.html b/build/emd_emw_ep.html index 0deb446..fca0c76 100644 --- a/build/emd_emw_ep.html +++ b/build/emd_emw_ep.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave, \] or more succinctly:

-
+

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

Similary, we get the

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_emw_mpw.html b/build/emd_emw_mpw.html index 2995cae..fa31848 100644 --- a/build/emd_emw_mpw.html +++ b/build/emd_emw_mpw.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1629,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. Generalizing to propagation in the direction of an arbitrary wavevector \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emd_emw_we.html b/build/emd_emw_we.html index 7b322b5..c97e01e 100644 --- a/build/emd_emw_we.html +++ b/build/emd_emw_we.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1626,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \( Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), we get the

-
+

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

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emdm.html b/build/emdm.html index 95cd6ff..3149315 100644 --- a/build/emdm.html +++ b/build/emdm.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

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Created: 2022-02-10 Thu 08:32

diff --git a/build/emdm_Me.html b/build/emdm_Me.html index 5765049..cc39ac5 100644 --- a/build/emdm_Me.html +++ b/build/emdm_Me.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1620,7 +1620,7 @@ target="_blank">Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

-

Created: 2022-02-09 Wed 22:40

+

Created: 2022-02-10 Thu 08:32

diff --git a/build/emdm_Me_Mem.html b/build/emdm_Me_Mem.html index 81aeab8..a01206c 100644 --- a/build/emdm_Me_Mem.html +++ b/build/emdm_Me_Mem.html @@ -1,7 +1,7 @@ - + Pre-Quantum Electrodynamics @@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia \] We therefore have the

-
+

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

-