Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+47
-273
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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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@@ -426,26 +380,9 @@ Table of contents
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<a href="./ems_es_ef_cE.html#ems_es_ef_cE">The Curl of \({\bf E}\)</a><span class="headline-id">ems.es.ef.cE</span>
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</li>
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<li>
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<details open="">
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<summary class="toc-currentpage">
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<li class="toc-currentpage">
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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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@@ -1868,7 +1612,7 @@ Table of contents
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</ul>
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</details>
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</nav>
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<ul class="navigation-links"><li>Prev: <a href="ems_es_ef_cE.html">The Curl of \({\bf E}\) <small>[ems.es.ef.cE]</small></a></li><li>Next: <a href="ems_es_ef_Gl_fl.html">Field Lines, Flux and Gauss's Law <small>[ems.es.ef.Gl.fl]</small></a></li><li>Up: <a href="ems_es_ef.html">Electrostatic Fields <small>[ems.es.ef]</small></a></li></ul><div id="outline-container-ems_es_ef_Gl" class="outline-5">
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<ul class="navigation-links"><li>Prev: <a href="ems_es_ef_cE.html">The Curl of \({\bf E}\) <small>[ems.es.ef.cE]</small></a></li><li>Next: <a href="ems_es_ep.html">The Electrostatic Potential <small>[ems.es.ep]</small></a></li><li>Up: <a href="ems_es_ef.html">Electrostatic Fields <small>[ems.es.ef]</small></a></li></ul><div id="outline-container-ems_es_ef_Gl" class="outline-5">
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||||
<h5 id="ems_es_ef_Gl">Gauss's Law: the divergence of \({\bf E}\)<a class="headline-permalink" href="./ems_es_ef_Gl.html#ems_es_ef_Gl"><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"/>
|
||||
<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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@@ -1911,7 +1655,7 @@ sphere of radius \(r\) around the charge,
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<p>
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\[
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\oint {\bf E} \cdot d{\bf a} = \frac{1}{4\pi\varepsilon_0} \int_{\cal S} \frac{q}{r^2} \hat{\bf r} \cdot \hat{\bf r}
|
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r^2 \sin \theta d\theta d\phi = \frac{q}{\varepsilon_0}
|
||||
~r^2 \sin \theta d\theta d\phi = \frac{q}{\varepsilon_0}
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\label{Gr(2.12)}
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||||
\]
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||||
</p>
|
||||
@@ -1919,7 +1663,22 @@ r^2 \sin \theta d\theta d\phi = \frac{q}{\varepsilon_0}
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<p>
|
||||
so by superposition, we obtain
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||||
</p>
|
||||
<div class="core div" id="org530a1c6">
|
||||
<div class="eqlabel" id="org567fae3">
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||||
<p>
|
||||
<a id="Gl_i"></a><a href="./ems_es_ef_Gl.html#Gl_i"><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>
|
||||
</p>
|
||||
<div class="alteqlabels" id="orgd08d994">
|
||||
<ul class="org-ul">
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||||
<li>Gr (2.13)</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="org107f70e">
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||||
<p>
|
||||
<b>Gauss' law (in integral form)</b>
|
||||
</p>
|
||||
@@ -1927,7 +1686,7 @@ so by superposition, we obtain
|
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<p>
|
||||
\[
|
||||
\oint_{\cal S} {\bf E} \cdot d{\bf a} = \frac{Q_{\mbox{enc}}}{\varepsilon_0}
|
||||
\label{Gr(2.13)}
|
||||
\tag{Gl_i}\label{Gl_i}
|
||||
\]
|
||||
</p>
|
||||
|
||||
@@ -1951,7 +1710,22 @@ By applying the divergence theorem,
|
||||
and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume
|
||||
is arbitrary, we get
|
||||
</p>
|
||||
<div class="core div" id="org1a1cc0c">
|
||||
<div class="eqlabel" id="orgcbc0cba">
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||||
<p>
|
||||
<a id="Gl_d"></a><a href="./ems_es_ef_Gl.html#Gl_d"><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"/>
|
||||
</svg></a>
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||||
</p>
|
||||
<div class="alteqlabels" id="org4033327">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (2.14)</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="core div" id="orgb66b8f3">
|
||||
<p>
|
||||
<b>Gauss' law in differential form</b>
|
||||
</p>
|
||||
@@ -1959,14 +1733,14 @@ is arbitrary, we get
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}.
|
||||
\label{Gr(2.14)}
|
||||
\tag{Gl_d}\label{Gl_d}
|
||||
\]
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We can also compute the divergence of \({\bf E}\) directly from \ref{eq:E_from_rho}:
|
||||
We can also compute the divergence of \({\bf E}\) directly from <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>:
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1976,7 +1750,7 @@ We can also compute the divergence of \({\bf E}\) directly from \ref{eq:E_from_r
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Using \ref{Gr(1.100)},
|
||||
Using <a href="./c_m_dd_3d.html#divdel">divdel</a>,
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -2010,7 +1784,7 @@ cylindrical or plane symmetry.
|
||||
Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="orge9fd2d4">
|
||||
<div class="example div" id="org34f5bc3">
|
||||
<p>
|
||||
<b>Example 2.2</b>: Field outside a uniformly charged sphere of radius \(R\) and total charge \(q\).
|
||||
</p>
|
||||
@@ -2041,7 +1815,7 @@ Same as point charge at origin!
|
||||
|
||||
</div>
|
||||
|
||||
<div class="example div" id="org53e8ed3">
|
||||
<div class="example div" id="org588cdee">
|
||||
<p>
|
||||
<b>Example 2.3</b>: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.
|
||||
</p>
|
||||
@@ -2080,7 +1854,7 @@ Therefore,
|
||||
</div>
|
||||
|
||||
|
||||
<div class="example div" id="org241823e">
|
||||
<div class="example div" id="orgd319ae5">
|
||||
<p>
|
||||
<b>Example 2.4</b>: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).
|
||||
</p>
|
||||
@@ -2106,7 +1880,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
|
||||
|
||||
</div>
|
||||
|
||||
<div class="example div" id="orgb7d665f">
|
||||
<div class="example div" id="org65c3abd">
|
||||
<p>
|
||||
<b>Example 2.5</b>: two infinite planes (put them vertical) carrying equal but opposite uniform surface charge densities \(\pm \sigma\).
|
||||
</p>
|
||||
@@ -2123,7 +1897,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
|
||||
|
||||
<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