Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+15
-271
@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-07 Mon 08:02 -->
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<!-- 2022-02-08 Tue 06:55 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -348,58 +348,12 @@ Table of contents
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>
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</summary>
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<ul>
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<li>
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<a href="#org788e483">Before Coulomb</a>
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</li>
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<li>
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<a href="#org8b037d5">Cavendish's experiment</a>
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</li>
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<li>
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<a href="#org1f82edc">Coulomb</a>
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</li>
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<li>
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<a href="#org359fd13">Current status</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_efo_e.html#ems_es_efo_e">Energy in Systems of Point Charges</a><span class="headline-id">ems.es.efo.e</span>
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</summary>
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<ul>
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<li>
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<a href="#ems_es_efo_e_p">Work; Pairwise Energy</a>
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</li>
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<li>
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<a href="#ems_es_efo_e_ga">Generic assembly</a>
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</li>
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<li>
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<a href="#ems_es_efo_e_cl">Crystal lattices</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -427,25 +381,8 @@ Table of contents
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_ef_Gl.html#ems_es_ef_Gl">Gauss's Law: the divergence of \({\bf E}\)</a><span class="headline-id">ems.es.ef.Gl</span>
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</summary>
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<ul>
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<li>
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<a href="#ems_es_ef_Gl_fl">Field Lines, Flux and Gauss's Law</a>
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</li>
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<li>
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<a href="#ems_es_ef_Gl_ex">Examples of applications of Gauss's law</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -455,7 +392,7 @@ Table of contents
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<details>
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<summary>
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<a href="./ems_es_ep.html#ems_es_ep">Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
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<a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
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</summary>
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@@ -492,7 +429,7 @@ Table of contents
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<details>
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<summary>
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<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy</a><span class="headline-id">ems.es.e</span>
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<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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</summary>
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@@ -565,29 +502,8 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>
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</summary>
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<ul>
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<li>
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<a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a>
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</li>
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<li>
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<a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a>
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</li>
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<li>
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<a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<a href="./ems_ca_fe_g.html#ems_ca_fe_g">Green's Identities</a><span class="headline-id">ems.ca.fe.g</span>
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@@ -871,33 +787,8 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./emsm_esm_di_ld.html#emsm_esm_di_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span>
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</summary>
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<ul>
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<li>
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<a href="#emsm_esm_d_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a>
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</li>
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<li>
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<a href="#emsm_esm_di_ld_bvp">Boundary Value Problems with Linear Dielectrics</a>
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</li>
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<li>
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<a href="#emsm_esm_di_ld_e">Energy in Dielectric Systems</a>
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</li>
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<li>
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<a href="#emsm_esm_di_ld_f">Forces on Dielectrics</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -926,21 +817,8 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>
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</summary>
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<ul>
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<li>
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<a href="#org65874b3">Why is Ferromagnetism such an intriguing phenomenon?</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<a href="./emsm_msm_m_fdi.html#emsm_msm_m_fdi">Torques and Forces on Magnetic Dipoles</a><span class="headline-id">emsm.msm.m.fdi</span>
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@@ -989,25 +867,8 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./emsm_msm_H_A.html#emsm_msm_H_A">Ampère's Law in Magnetized Materials</a><span class="headline-id">emsm.msm.H.A</span>
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</summary>
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<ul>
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<li>
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<a href="#emsm_msm_H_A_dp">A Deceptive Parallel</a>
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</li>
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<li>
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<a href="#emsm_msm_H_A_elm">Energy in Linear Media</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -1599,37 +1460,8 @@ Table of contents
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</li>
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<li>
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<details>
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<summary>
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<a href="./c_m_dc_d2.html#c_m_dc_d2">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span>
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</summary>
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<ul>
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<li>
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<a href="#orge025182">Divergence of gradient</a>
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</li>
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<li>
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<a href="#orgacb930d">Curl of a gradient</a>
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</li>
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<li>
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<a href="#org6caee98">Gradient of the divergence</a>
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</li>
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<li>
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<a href="#orgb5da747">Divergence of a curl</a>
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</li>
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<li>
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<a href="#orgebcbadc">Curl of curl</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -1645,29 +1477,8 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./c_m_ic_lsv.html#c_m_ic_lsv">Line, Surface and Volume Integrals</a><span class="headline-id">c.m.ic.lsv</span>
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</summary>
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<ul>
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<li>
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<a href="#org638a76f">Line Integrals</a>
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</li>
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<li>
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<a href="#orgd8e925a">Surface Integrals</a>
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</li>
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<li>
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<a href="#org5c24b4a">Volume Integrals</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<a href="./c_m_ic_ftc.html#c_m_ic_ftc">The Fundamental Theorem of Calculus</a><span class="headline-id">c.m.ic.ftc</span>
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@@ -1703,62 +1514,12 @@ Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>
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</summary>
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<ul>
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<li>
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<a href="#c_m_cs_sph_grad">Gradient</a>
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</li>
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<li>
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<a href="#c_m_cs_sph_div">Divergence</a>
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</li>
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<li>
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<a href="#c_m_cs_sph_curl">Curl</a>
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</li>
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<li>
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<a href="#c_m_cs_sph_lap">Laplacian</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>
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</summary>
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<ul>
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<li>
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<a href="#c_m_cs_cyl_grad">Gradient</a>
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</li>
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<li>
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<a href="#c_m_cs_cyl_div">Divergence</a>
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</li>
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<li>
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<a href="#c_m_cs_cyl_curl">Curl</a>
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</li>
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<li>
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<a href="#c_m_cs_cyl_lap">Laplacian</a>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>
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@@ -1807,25 +1568,8 @@ Table of contents
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</li>
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<li>
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<details>
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<summary>
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<a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>
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</summary>
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<ul>
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<li>
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<a href="#c_m_vf_pot_irrot">Theorem 1: Curl-less (irrotational) fields</a>
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</li>
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<li>
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<a href="#c_m_vf_pot_solen">Theorem 2: Divergence-less (solenoidal) fields</a>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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@@ -1910,7 +1654,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
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\]
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We therefore have the
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</p>
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<div class="core div" id="orgb5c72da">
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<div class="core div" id="orgb555ebf">
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<p>
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{\bf Polarization current density}
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\[
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@@ -1928,7 +1672,7 @@ the polarization current is the result of linear motion of charge when
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polarization changes). We can check consistency with the continuity equation
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associated to the conservation of bound charges:
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</p>
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<aside id="org7ce3b70">
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<aside id="org86c5ff2">
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<p>
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Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
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\({\boldsymbol J}_b\) but this is not the convention used here.
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@@ -1951,7 +1695,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
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In view of this: total charge density can be separated into 2 parts,
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{\it free} and {\it bound}:
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</p>
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<div class="main div" id="org3b2083c">
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<div class="main div" id="orgf8c82b2">
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<p>
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\[
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\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
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@@ -1964,10 +1708,10 @@ In view of this: total charge density can be separated into 2 parts,
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and current can be separated into three parts, {\it free}, {\it bound} and
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{\it polarization}:
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</p>
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<div class="main div" id="org09ed727">
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<div class="main div" id="org4054380">
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<p>
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\[
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{\bf J} = {\bf J}<sub>f</sub> + {\bf J}<sub>b</sub> + {\bf J}<sub>p</sub> = {\bf J}<sub>f</sub> + {\boldsymbol ∇} × {\bf M}
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{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
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</p>
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<ul class="org-ul">
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<li>\frac{∂ {\bf P}}{∂ t}.</li>
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@@ -1988,7 +1732,7 @@ Gauss's law: can be rewritten
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\]
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where (as in static case)
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</p>
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<div class="core div" id="org508fb0a">
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<div class="core div" id="orgf78b92c">
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<p>
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\[
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{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
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@@ -2000,10 +1744,10 @@ where (as in static case)
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<p>
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Ampère's law including Maxwell's term:
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\[
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{\boldsymbol ∇} × {\bf B} = μ<sub>0</sub> \left( {\bf J}<sub>f</sub> + {\boldsymbol ∇} × {\bf M}
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{\boldsymbol ∇} × {\bf B} = μ_0 \left( {\bf J}_f + {\boldsymbol ∇} × {\bf M}
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</p>
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<ul class="org-ul">
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<li>\frac{∂ {\bf P}}{∂ t} \right) + μ<sub>0</sub> ε<sub>0</sub> \frac{∂ {\bf E}}{∂ t},</li>
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<li>\frac{∂ {\bf P}}{∂ t} \right) + μ_0 ε_0 \frac{∂ {\bf E}}{∂ t},</li>
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</ul>
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<p>
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\]
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@@ -2014,7 +1758,7 @@ or
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\]
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where as before
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</p>
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<div class="core div" id="org6879249">
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<div class="core div" id="org513c514">
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<p>
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\[
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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@@ -2032,7 +1776,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
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<p>
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In terms of free charges and currents, we thus get
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</p>
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<div class="core div" id="orgb7d4703">
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<div class="core div" id="org686ea33">
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<p>
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{\bf Maxwell's equations {\it (in matter)}}
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</p>
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@@ -2058,7 +1802,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
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in terms of \({\bf E}\) and \({\bf B}\).
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For the restricted case of linear media:
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</p>
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<div class="main div" id="org30a7942">
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<div class="main div" id="org6accc1d">
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<p>
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\[
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{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
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@@ -2082,7 +1826,7 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
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<hr><div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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||||
<p class="date">Created: 2022-02-07 Mon 08:02</p>
|
||||
<p class="date">Created: 2022-02-08 Tue 06:55</p>
|
||||
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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||||
</div>
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||||
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