Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+15
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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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@@ -1885,7 +1629,7 @@ For a single dipole: refer to \ref{Gr(5.83)} (vector potential of single dipole
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For a chunk of material with local magnetization \({\bf M} ({\bf r})\),
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by the principle of superposition we thus have:
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</p>
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<div class="main div" id="orgdea3ffd">
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<div class="main div" id="orgcc63d5d">
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<p>
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\[
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{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \int_{\cal V} d\tau' ~\frac{{\bf M} ({\bf r}') \times ({\bf r} - {\bf r}')}{|{\bf r} - {\bf r}'|^3}
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@@ -1906,10 +1650,10 @@ using \({\boldsymbol \nabla}' \frac{1}{|{\bf r} - {\bf r}'|} = \frac{{\bf r} - {
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we can further integrate by parts and use product rule: \({\boldsymbol \nabla} \times (f {\bf A}) = f ( {\boldsymbol \nabla} \times {\bf A}) - {\bf A} \times ({\boldsymbol \nabla} f)\), we get
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\[
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{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} \left\{
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∫<sub>\cal V</sub> dτ' \frac{{\boldsymbol ∇}' × {\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|}
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∫_{\cal V} dτ' \frac{{\boldsymbol ∇}' × {\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|}
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</p>
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<ul class="org-ul">
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<li>∫<sub>\cal V</sub> dτ' {\boldsymbol ∇}' × \left( \frac{{\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|} \right)</li>
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<li>∫_{\cal V} dτ' {\boldsymbol ∇}' × \left( \frac{{\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|} \right)</li>
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</ul>
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<p>
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\right\}
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@@ -1917,17 +1661,17 @@ we can further integrate by parts and use product rule: \({\boldsymbol \nabla} \
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Problem 1.61 b) (p.56): leads to
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\[
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{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi}
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∫<sub>\cal V</sub> dτ' \frac{{\boldsymbol ∇}' × {\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|}
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∫_{\cal V} dτ' \frac{{\boldsymbol ∇}' × {\bf M} ({\bf r}')}{|{\bf r} - {\bf r}'|}
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</p>
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<ul class="org-ul">
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<li>\frac{\mu_0}{4\pi} \oint<sub>\cal S</sub> \frac{{\bf M} ({\bf r}') × d{\bf a}'}{|{\bf r} - {\bf r}'|}</li>
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<li>\frac{\mu_0}{4\pi} \oint_{\cal S} \frac{{\bf M} ({\bf r}') × d{\bf a}'}{|{\bf r} - {\bf r}'|}</li>
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</ul>
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<p>
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\label{Gr(6.12)}
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\]
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Reinterpretation: first term: potential from volume current,
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</p>
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<div class="main div" id="org9491d55">
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<div class="main div" id="org2a43517">
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<p>
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\[
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{\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
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@@ -1939,7 +1683,7 @@ Reinterpretation: first term: potential from volume current,
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<p>
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second term: potential from surface current,
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</p>
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<div class="main div" id="orgaefe466">
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<div class="main div" id="org78e636c">
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<p>
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\[
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{\bf K}_b = {\bf M} \times \hat{\bf n}
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@@ -1951,13 +1695,13 @@ second term: potential from surface current,
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<p>
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With these definitions,
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</p>
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<div class="main div" id="org64d01be">
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<div class="main div" id="org258eef0">
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<p>
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\[
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{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} ∫<sub>\cal V</sub> dτ' \frac{{\bf J}<sub>b</sub> ({\bf r}')}{|{\bf r} - {\bf r}'|}
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{\bf A} ({\bf r}) = \frac{\mu_0}{4\pi} ∫_{\cal V} dτ' \frac{{\bf J}_b ({\bf r}')}{|{\bf r} - {\bf r}'|}
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</p>
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<ul class="org-ul">
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<li>\frac{\mu_0}{4\pi} \oint<sub>\cal S</sub> da' \frac{{\bf K}<sub>b</sub> ({\bf r}')}{|{\bf r} - {\bf r}'|}</li>
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<li>\frac{\mu_0}{4\pi} \oint_{\cal S} da' \frac{{\bf K}_b ({\bf r}')}{|{\bf r} - {\bf r}'|}</li>
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</ul>
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<p>
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\label{Gr(6.15)}
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@@ -1972,7 +1716,7 @@ in the volume and surface of the material.
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<div class="example div" id="org1b7c642">
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<div class="example div" id="org167263a">
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<p>
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\paragraph{Example 6.1:} find field of uniformly magnetized sphere.
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\paragraph{Solution:} put z axis along \({\bf M}\).
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@@ -2005,7 +1749,7 @@ inside sphere, whereas outside: pure dipole field, with
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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>
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<p class="date">Created: 2022-02-08 Tue 06:55</p>
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||||
<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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