Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+43
-269
@@ -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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||||
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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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||||
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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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|
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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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||||
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||||
</ul>
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||||
</details>
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</li>
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</ul>
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@@ -1884,11 +1628,26 @@ the <b>electrostatic field</b>.
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<p>
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Going back to our source and test charge setup, we shall thus think of the
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force on the test charge as being given by its coupling to
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an electric field originating from the source charge. Invoking the
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superposition principle, we can thus write
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an electric field originating from the source charge.
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Invoking the superposition principle, we can thus write
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</p>
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<div class="core div" id="org485b6df">
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<div class="eqlabel" id="org2b84014">
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<p>
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<a id="FqE"></a><a href="./ems_es_ef_pc.html#FqE"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org0dcd3af">
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<ul class="org-ul">
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<li>Gr (2.3)</li>
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</ul>
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</div>
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</div>
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<div class="core div" id="org8d0792d">
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<p>
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</p>
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@@ -1896,15 +1655,30 @@ superposition principle, we can thus write
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<p>
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\[
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{\bf F}_t = q_t {\bf E}({\bf r}_t)
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\label{Gr(2.3)}
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\tag{FqE}\label{FqE}
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\]
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</p>
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</div>
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<p>
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where
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with the electric field of a point charge distribution being
|
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</p>
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<div class="core div" id="org28ba57b">
|
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<div class="eqlabel" id="orgd7a36a0">
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<p>
|
||||
<a id="E_pcd"></a><a href="./ems_es_ef_pc.html#E_pcd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
|
||||
<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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||||
<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
|
||||
</svg></a>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org5d572cf">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (2.4)</li>
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||||
</ul>
|
||||
|
||||
</div>
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||||
|
||||
</div>
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||||
<div class="core div" id="org9d51306">
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<p>
|
||||
|
||||
</p>
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@@ -1912,7 +1686,7 @@ where
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||||
<p>
|
||||
\[
|
||||
{\bf E} ({\bf r}) \equiv \frac{1}{4\pi \varepsilon_0} \sum_{i=1}^n q_i \frac{{\bf r} - {\bf r}_i}{|{\bf r} - {\bf r}_i|^3}
|
||||
\label{Gr(2.4)}
|
||||
\tag{E_pcd}\label{E_pcd}
|
||||
\]
|
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</p>
|
||||
|
||||
@@ -1922,7 +1696,7 @@ The electric field \({\bf E} ({\bf r})\) is thus the force per unit charge that
|
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be exerted if you put a test charge at position \({\bf r}\).
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org7096351">
|
||||
<div class="example div" id="org116584f">
|
||||
<p>
|
||||
<b>Example</b>
|
||||
</p>
|
||||
@@ -1939,7 +1713,7 @@ We have \(|{\bf r} - {\bf r}'| = \sqrt{z^2 + (d/2)^2}\) for both charges and \(\
|
||||
|
||||
<p>
|
||||
\[
|
||||
{\bf E} = \frac{1}{4\pi \varepsilon_0} \frac{2q z}{|{\bf r} - {\bf r}'|^{3/2}} \hat{\bf z}
|
||||
{\bf E} = \frac{1}{4\pi \varepsilon_0} \frac{2q z}{[z^2 + d^2/4]^{3/2}} ~\hat{\bf z}
|
||||
\]
|
||||
</p>
|
||||
|
||||
@@ -1953,7 +1727,7 @@ We have \(|{\bf r} - {\bf r}'| = \sqrt{z^2 + (d/2)^2}\) for both charges and \(\
|
||||
|
||||
<hr><div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<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>
|
||||
</div>
|
||||
|
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Reference in New Issue
Block a user