Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+23
-279
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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>
|
||||
<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 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_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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||||
|
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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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||||
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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>
|
||||
<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>
|
||||
<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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@@ -1868,20 +1612,26 @@ 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="org359fd13.html">Current status <small>[org359fd13]</small></a></li><li>Next: <a href="ems_es_efo_e_p.html">Work; Pairwise Energy <small>[ems.es.efo.e.p]</small></a></li><li>Up: <a href="ems_es_efo.html">Electric Force and Energy <small>[ems.es.efo]</small></a></li></ul><div id="outline-container-ems_es_efo_e" class="outline-5">
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||||
<ul class="navigation-links"><li>Prev: <a href="ems_es_efo_exp.html">Experimental Investigations <small>[ems.es.efo.exp]</small></a></li><li>Next: <a href="ems_es_ef.html">Electrostatic Fields <small>[ems.es.ef]</small></a></li><li>Up: <a href="ems_es_efo.html">Electric Force and Energy <small>[ems.es.efo]</small></a></li></ul><div id="outline-container-ems_es_efo_e" class="outline-5">
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||||
<h5 id="ems_es_efo_e">Energy in Systems of Point Charges<a class="headline-permalink" href="./ems_es_efo_e.html#ems_es_efo_e"><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><span class="headline-id">ems.es.efo.e</span></h5>
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||||
<div class="outline-text-5" id="text-ems_es_efo_e">
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||||
<ul class="org-ul altsecnrs">
|
||||
<li>FLS II 8</li>
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||||
<li>Gr 2.4.2</li>
|
||||
<li>PM 1.5</li>
|
||||
</ul>
|
||||
</div>
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||||
|
||||
<div id="outline-container-ems_es_efo_e_p" class="outline-6">
|
||||
<h6 id="ems_es_efo_e_p"><a href="#ems_es_efo_e_p">Work; Pairwise Energy</a></h6>
|
||||
<div class="outline-text-6" id="text-ems_es_efo_e_p">
|
||||
<p>
|
||||
Consider some distribution of charge which produces an electric field.
|
||||
How much work is needed to carry a small test charge \(q_t\) from one place to another, <i>i.e.</i> from point \({\bf a}\) to point \({\bf b}\) ?
|
||||
Consider some distribution of charge which is held statically in place.
|
||||
How much work is needed to adiabatically move a small test charge \(q_t\) from one
|
||||
position to another, <i>i.e.</i> from point \({\bf a}\) to point \({\bf b}\) ?
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1911,8 +1661,8 @@ For the path, the 'angular' part is not contributing (see drawings in
|
||||
FLS II 4-3). The integral is thus purely radial,
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="orgdf595d6">
|
||||
<div class="alteqlabels" id="org6cae870">
|
||||
<div class="eqlabel" id="org70d5df1">
|
||||
<div class="alteqlabels" id="org206a74a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (2.18)</li>
|
||||
</ul>
|
||||
@@ -1929,14 +1679,14 @@ FLS II 4-3). The integral is thus purely radial,
|
||||
<p>
|
||||
As should be clear by now, this result does not depend on the path
|
||||
(if it did, we'd have a <i>perpetuum mobile</i> when going from \({\bf r}_a\)
|
||||
to \({\bf r}_a\) one way, and coming back another).
|
||||
to \({\bf r}_b\) one way, and coming back another).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
When thinking about the energy of this pair of charges, we think
|
||||
of starting from an initial configuration where the charges are infinitely
|
||||
distance (which we associate to zero energy), and thus set \({\bf r}_a = \infty\).
|
||||
We can thus write
|
||||
For a pair of charges, the energy is thus
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@@ -1944,17 +1694,10 @@ We can thus write
|
||||
W = \frac{1}{4\pi \varepsilon_0} \frac{q_t q_s}{|{\bf r}_t - {\bf r}_s|}
|
||||
\]
|
||||
</p>
|
||||
|
||||
|
||||
<div class="example div" id="orgfdafc11">
|
||||
<p>
|
||||
<b>Example</b>: fission
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-ems_es_efo_e_ga" class="outline-6">
|
||||
<h6 id="ems_es_efo_e_ga"><a href="#ems_es_efo_e_ga">Generic assembly</a></h6>
|
||||
<div class="outline-text-6" id="text-ems_es_efo_e_ga">
|
||||
@@ -1982,9 +1725,10 @@ where in the second equality we have symmetrized the sums for convenience.
|
||||
<div id="outline-container-ems_es_efo_e_cl" class="outline-6">
|
||||
<h6 id="ems_es_efo_e_cl"><a href="#ems_es_efo_e_cl">Crystal lattices</a></h6>
|
||||
<div class="outline-text-6" id="text-ems_es_efo_e_cl">
|
||||
<p>
|
||||
PM3:1.6
|
||||
</p>
|
||||
<ul class="org-ul altsecnrs">
|
||||
<li>FLS II 8-4</li>
|
||||
<li>PM 1.6</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1992,7 +1736,7 @@ PM3:1.6
|
||||
|
||||
<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