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2022-02-08 07:07:41 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-07 Mon 08:02 -->
<!-- 2022-02-08 Tue 06:55 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -348,58 +348,12 @@ Table of contents
</li>
<li>
<details>
<summary>
<a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>
</summary>
<ul>
<li>
<a href="#org788e483">Before Coulomb</a>
</li>
<li>
<a href="#org8b037d5">Cavendish's experiment</a>
</li>
<li>
<a href="#org1f82edc">Coulomb</a>
</li>
<li>
<a href="#org359fd13">Current status</a>
</li>
</ul>
</details>
</li>
<li>
<details>
<summary>
<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>
</summary>
<ul>
<li>
<a href="#ems_es_efo_e_p">Work; Pairwise Energy</a>
</li>
<li>
<a href="#ems_es_efo_e_ga">Generic assembly</a>
</li>
<li>
<a href="#ems_es_efo_e_cl">Crystal lattices</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -427,25 +381,8 @@ Table of contents
</li>
<li>
<details>
<summary>
<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>
</summary>
<ul>
<li>
<a href="#ems_es_ef_Gl_fl">Field Lines, Flux and Gauss's Law</a>
</li>
<li>
<a href="#ems_es_ef_Gl_ex">Examples of applications of Gauss's law</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -455,7 +392,7 @@ Table of contents
<details>
<summary>
<a href="./ems_es_ep.html#ems_es_ep">Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
<a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
</summary>
@@ -492,7 +429,7 @@ Table of contents
<details>
<summary>
<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy</a><span class="headline-id">ems.es.e</span>
<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
</summary>
@@ -565,29 +502,8 @@ Table of contents
</summary>
<ul>
<li>
<details>
<summary>
<a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>
</summary>
<ul>
<li>
<a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a>
</li>
<li>
<a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a>
</li>
<li>
<a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a>
</li>
</ul>
</details>
</li>
<li>
<a href="./ems_ca_fe_g.html#ems_ca_fe_g">Green's Identities</a><span class="headline-id">ems.ca.fe.g</span>
@@ -870,34 +786,9 @@ Table of contents
</summary>
<ul>
<li>
<details open="">
<summary class="toc-currentpage">
<li class="toc-currentpage">
<a href="./emsm_esm_di_ld.html#emsm_esm_di_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span>
</summary>
<ul>
<li>
<a href="#emsm_esm_d_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a>
</li>
<li>
<a href="#emsm_esm_di_ld_bvp">Boundary Value Problems with Linear Dielectrics</a>
</li>
<li>
<a href="#emsm_esm_di_ld_e">Energy in Dielectric Systems</a>
</li>
<li>
<a href="#emsm_esm_di_ld_f">Forces on Dielectrics</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -926,21 +817,8 @@ Table of contents
</summary>
<ul>
<li>
<details>
<summary>
<a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>
</summary>
<ul>
<li>
<a href="#org65874b3">Why is Ferromagnetism such an intriguing phenomenon?</a>
</li>
</ul>
</details>
</li>
<li>
<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>
@@ -989,25 +867,8 @@ Table of contents
</summary>
<ul>
<li>
<details>
<summary>
<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>
</summary>
<ul>
<li>
<a href="#emsm_msm_H_A_dp">A Deceptive Parallel</a>
</li>
<li>
<a href="#emsm_msm_H_A_elm">Energy in Linear Media</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -1599,37 +1460,8 @@ Table of contents
</li>
<li>
<details>
<summary>
<a href="./c_m_dc_d2.html#c_m_dc_d2">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span>
</summary>
<ul>
<li>
<a href="#orge025182">Divergence of gradient</a>
</li>
<li>
<a href="#orgacb930d">Curl of a gradient</a>
</li>
<li>
<a href="#org6caee98">Gradient of the divergence</a>
</li>
<li>
<a href="#orgb5da747">Divergence of a curl</a>
</li>
<li>
<a href="#orgebcbadc">Curl of curl</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -1645,29 +1477,8 @@ Table of contents
</summary>
<ul>
<li>
<details>
<summary>
<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>
</summary>
<ul>
<li>
<a href="#org638a76f">Line Integrals</a>
</li>
<li>
<a href="#orgd8e925a">Surface Integrals</a>
</li>
<li>
<a href="#org5c24b4a">Volume Integrals</a>
</li>
</ul>
</details>
</li>
<li>
<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>
@@ -1703,62 +1514,12 @@ Table of contents
</summary>
<ul>
<li>
<details>
<summary>
<a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>
</summary>
<ul>
<li>
<a href="#c_m_cs_sph_grad">Gradient</a>
</li>
<li>
<a href="#c_m_cs_sph_div">Divergence</a>
</li>
<li>
<a href="#c_m_cs_sph_curl">Curl</a>
</li>
<li>
<a href="#c_m_cs_sph_lap">Laplacian</a>
</li>
</ul>
</details>
</li>
<li>
<details>
<summary>
<a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>
</summary>
<ul>
<li>
<a href="#c_m_cs_cyl_grad">Gradient</a>
</li>
<li>
<a href="#c_m_cs_cyl_div">Divergence</a>
</li>
<li>
<a href="#c_m_cs_cyl_curl">Curl</a>
</li>
<li>
<a href="#c_m_cs_cyl_lap">Laplacian</a>
</li>
</ul>
</details>
</li>
<li>
<a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>
@@ -1807,25 +1568,8 @@ Table of contents
</li>
<li>
<details>
<summary>
<a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>
</summary>
<ul>
<li>
<a href="#c_m_vf_pot_irrot">Theorem 1: Curl-less (irrotational) fields</a>
</li>
<li>
<a href="#c_m_vf_pot_solen">Theorem 2: Divergence-less (solenoidal) fields</a>
</li>
</ul>
</details>
</li>
</ul>
@@ -1868,7 +1612,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_di.html">Dielectrics&emsp;<small>[emsm.esm.di]</small></a></li><li>Next:&nbsp;<a href="emsm_esm_d_ld_sp.html">Susceptibility, Permittivity, Dielectric Constant&emsp;<small>[emsm.esm.d.ld.sp]</small></a></li><li>Up:&nbsp;<a href="emsm_esm_di.html">Dielectrics&emsp;<small>[emsm.esm.di]</small></a></li></ul><div id="outline-container-emsm_esm_di_ld" class="outline-5">
<ul class="navigation-links"><li>Prev:&nbsp;<a href="emsm_esm_di.html">Dielectrics&emsp;<small>[emsm.esm.di]</small></a></li><li>Next:&nbsp;<a href="emsm_msm.html">Magnetostatics in matter&emsp;<small>[emsm.msm]</small></a></li><li>Up:&nbsp;<a href="emsm_esm_di.html">Dielectrics&emsp;<small>[emsm.esm.di]</small></a></li></ul><div id="outline-container-emsm_esm_di_ld" class="outline-5">
<h5 id="emsm_esm_di_ld">Linear Dielectrics<a class="headline-permalink" href="./emsm_esm_di_ld.html#emsm_esm_di_ld"><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"/>
@@ -1882,7 +1626,7 @@ Table of contents
<p>
For many substances: polarization is proportional to field, if the latter isn't too strong:
</p>
<div class="main div" id="orgd58d8ec">
<div class="main div" id="org5882cee">
<p>
\[
{\bf P} = \varepsilon_0 \chi_e {\bf E}
@@ -1912,7 +1656,7 @@ In linear dielectrics:
\]
so
</p>
<div class="main div" id="org095e84e">
<div class="main div" id="orga3558f5">
<p>
\[
{\bf D} = \varepsilon {\bf E}
@@ -1934,7 +1678,7 @@ This is all just nomenclature, everything is already in \ref{Gr(4.30)}.
<div class="example div" id="org7c7bb44">
<div class="example div" id="org2c914e3">
<p>
\paragraph{Example 4.5:} metal sphere of radius \(a\) carrying charge \(Q\), surrounded out to radius \(b\) by
a linear dielectric material of permittivity \(\varepsilon\). Find potential at center (relative to infinity).
@@ -1990,7 +1734,7 @@ of {\it e.g.} \({\bf P}\) would not vanish.
Only case where parallel works: space entirely filled with homogeneous linear dielectric.
</p>
<div class="example div" id="org35ed17c">
<div class="example div" id="orgfebd47e">
<p>
\paragraph{Example 4.6:} parallel-plate capacitor filled with insulating material of
dielectric constant \(\varepsilon_r\). What is the effect on the capacitance ?
@@ -2025,7 +1769,7 @@ If \(\rho = 0\), any net charge is on surface, potential then obeys Laplace.
<p>
Convenient to rewrite boundary conditions in terms of free charge: from \ref{Gr(4.26)},
</p>
<div class="main div" id="orgf1c3d36">
<div class="main div" id="orgb605e6c">
<p>
\[
\varepsilon_{above} E^{\perp}_{above} - \varepsilon_{below} E^{\perp}_{below} = \sigma_f
@@ -2037,7 +1781,7 @@ Convenient to rewrite boundary conditions in terms of free charge: from \ref{Gr
<p>
or in terms of the potential,
</p>
<div class="main div" id="orgd8c244f">
<div class="main div" id="orgf35ca50">
<p>
\[
\varepsilon_{above} \frac{\partial V_{above}}{\partial n} -
@@ -2050,7 +1794,7 @@ or in terms of the potential,
<p>
Potential itself is continuous,
</p>
<div class="main div" id="org3f0f257">
<div class="main div" id="org5c553e0">
<p>
\[
V_{above} = V_{below}
@@ -2062,7 +1806,7 @@ Potential itself is continuous,
<div class="example div" id="org3273491">
<div class="example div" id="orgf527756">
<p>
\paragraph{Example 4.7:} sphere of homogeneous dielectric material in uniform electric field \({\bf E}_0\).
Find electric field inside sphere.
@@ -2120,7 +1864,7 @@ Thus,
<div class="example div" id="orgec0d73b">
<div class="example div" id="orgaa2beb6">
<p>
\paragraph{Example 4.8:} suppose region below \(z = 0\) is filled with uniform linear dielectric with susceptibility \(\chi_e\).
Calculate force on point charge \(q\) situated a distance \(d\) above origin.
@@ -2224,7 +1968,7 @@ Special case of linear isotropic dielectric: \({\bf D} = \varepsilon {\bf E}\),
\]
Total work done:
</p>
<div class="main div" id="orgf6db344">
<div class="main div" id="org9a3cd32">
<p>
\[
W = \frac{1}{2} \int d\tau {\bf D} \cdot {\bf E}
@@ -2315,7 +2059,7 @@ so like before but with the correct sign.
<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>
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