Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
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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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|
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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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<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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||||
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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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@@ -1880,23 +1624,22 @@ calculated from Coulomb's law using the superposition principle. Since each inf
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volume element \(d\tau' = dx' dy' dz'\) contains a charge \(dq' = \rho({\bf r}') d\tau'\), we have
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</p>
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<div class="eqlabel" id="org285c290">
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<div class="eqlabel" id="org93fd4f2">
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<p>
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||||
<a id="E_vcd"></a><a href="./ems_es_ef_ccd.html#E_vcd"><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="orgcdb6e74">
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||||
<div class="alteqlabels" id="org2fcc353">
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<ul class="org-ul">
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<li>Gr4(2.8)</li>
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<li>W3(??)</li>
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<li>Gr4 (2.8)</li>
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</ul>
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||||
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||||
</div>
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||||
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||||
</div>
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||||
<div class="main div" id="orge6f0db5">
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||||
<div class="main div" id="orgf2f415b">
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||||
<p>
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||||
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||||
</p>
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||||
@@ -1916,14 +1659,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
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\(\sigma({\bf r})\), we have over an infinitesimal area \(da'\) a charge \(dq' = \sigma({\bf r}') da'\), so
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</p>
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||||
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||||
<div class="eqlabel" id="org8d528cc">
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<div class="eqlabel" id="org6008079">
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||||
<p>
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||||
<a id="E_scd"></a><a href="./ems_es_ef_ccd.html#E_scd"><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="orge2b8781">
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||||
<div class="alteqlabels" id="orgc6962f2">
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||||
<ul class="org-ul">
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||||
<li>Gr4(2.7)</li>
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||||
</ul>
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@@ -1931,7 +1674,7 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
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</div>
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||||
|
||||
</div>
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||||
<div class="main div" id="org60c114f">
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||||
<div class="main div" id="org433e39b">
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||||
<p>
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||||
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||||
</p>
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||||
@@ -1948,14 +1691,14 @@ Similarly, if the charge is spread out over a two-dimensional surface \({\cal S}
|
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Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf r}')\),
|
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</p>
|
||||
|
||||
<div class="eqlabel" id="orgae4aadc">
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||||
<div class="eqlabel" id="orgca0563d">
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||||
<p>
|
||||
<a id="E_lcd"></a><a href="./ems_es_ef_ccd.html#E_lcd"><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"/>
|
||||
<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="orgc2b06f7">
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||||
<div class="alteqlabels" id="orgf23e73b">
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||||
<ul class="org-ul">
|
||||
<li>Gr (2.6)</li>
|
||||
</ul>
|
||||
@@ -1963,7 +1706,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
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||||
</div>
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||||
|
||||
</div>
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||||
<div class="main div" id="orgfa981b8">
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||||
<div class="main div" id="orgd91bb3b">
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||||
<p>
|
||||
|
||||
</p>
|
||||
@@ -1977,7 +1720,7 @@ Finally, for a line path \({\cal P}\) with linear charge density \(\lambda({\bf
|
||||
|
||||
</div>
|
||||
|
||||
<div class="example div" id="orgebf037d">
|
||||
<div class="example div" id="orgac43438">
|
||||
<p>
|
||||
<b>Example</b>
|
||||
</p>
|
||||
@@ -1992,8 +1735,8 @@ of length \(2L\) carrying a uniform line charge \(\lambda\).
|
||||
</p>
|
||||
|
||||
\begin{align*}
|
||||
{\bf r} &= z \hat{\bf z}, {\bf r}' = x \hat{\bf x}, dl' = dx,
|
||||
|{\bf r} - {\bf r}'| = \sqrt{z^2 + {x}^2},
|
||||
{\bf r} &= z \hat{\bf z}, {\bf r}' = x' \hat{\bf x}, dl' = dx',
|
||||
|{\bf r} - {\bf r}'| = \sqrt{z^2 + {x'}^2},
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
@@ -2001,7 +1744,7 @@ so we have
|
||||
</p>
|
||||
|
||||
\begin{align*}
|
||||
{\bf E} = \frac{1}{4\pi \varepsilon_0} \int_{-L}^L dx' \lambda \frac{z \hat{\bf z} - x \hat{\bf x}}{(z^2 + x^2)^{3/2}}
|
||||
{\bf E} = \frac{1}{4\pi \varepsilon_0} \int_{-L}^L dx' \lambda \frac{z \hat{\bf z} - x' \hat{\bf x}}{(z^2 + {x'}^2)^{3/2}}
|
||||
= \frac{\lambda}{4\pi \varepsilon_0} \left[ z \hat{\bf z} \int_{-L}^L dx \frac{1}{(z^2 + x^2)^{3/2}} - \hat{\bf x} \int_{-L}^L dx \frac{x}{(z^2 + x^2)^{3/2}} \right]
|
||||
\end{align*}
|
||||
|
||||
@@ -2011,7 +1754,7 @@ most easily by observing that \(\frac{d}{dx} \left( \frac{x}{\sqrt{z^2 + x^2}} \
|
||||
= \frac{1}{\sqrt{z^2 + x^2}} - \frac{x^2}{(z^2 + x^2)^{3/2}} = \frac{z^2}{(z^2 + x^2)^{3/2}}\),
|
||||
leading to
|
||||
</p>
|
||||
<aside id="org57b6ee2">
|
||||
<aside id="orgdec9553">
|
||||
<p>
|
||||
You could alternately proceed by using changes of variables \(y = zx\) followed by \(y = \tanh \alpha\):
|
||||
\(\int_{-L}^L \frac{dx}{(z^2 + x^2)^{3/2}} = \frac{1}{z^2}
|
||||
@@ -2055,7 +1798,7 @@ whereas for short distances \(z \ll L\) the field looks like that of an infinite
|
||||
|
||||
<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>
|
||||
|
||||
|
||||
Reference in New Issue
Block a user