Update 2022-02-08 07:07
This commit is contained in:
Jean-Sébastien
+13
-269
@@ -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>
|
||||
@@ -871,33 +787,8 @@ Table of contents
|
||||
</summary>
|
||||
<ul>
|
||||
<li>
|
||||
|
||||
<details>
|
||||
<summary>
|
||||
<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>
|
||||
@@ -1932,7 +1676,7 @@ But obviously,
|
||||
\]
|
||||
so we get
|
||||
\[
|
||||
{\bf E} ⋅ {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( ε<sub>0</sub> E<sup>2</sup> + \frac{1}{\mu_0} B<sup>2</sup> \right)
|
||||
{\bf E} ⋅ {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\frac{1}{\mu_0} {\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).</li>
|
||||
@@ -1943,14 +1687,14 @@ so we get
|
||||
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
|
||||
we obtain
|
||||
</p>
|
||||
<div class="main div" id="org27d4e56">
|
||||
<div class="main div" id="org4fee5c1">
|
||||
<p>
|
||||
{\bf Poynting's theorem}
|
||||
\[
|
||||
\frac{dW}{dt} = -\frac{d}{d t} ∫<sub>\cal V</sub> dτ \frac{1}{2} \left( ε<sub>0</sub> E<sup>2</sup> + \frac{1}{\mu_0} B<sup>2</sup> \right)
|
||||
\frac{dW}{dt} = -\frac{d}{d t} ∫_{\cal V} dτ \frac{1}{2} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\frac{1}{\mu_0} \oint<sub>\cal S</sub> d{\bf a} ⋅ ({\bf E} × {\bf B})</li>
|
||||
<li>\frac{1}{\mu_0} \oint_{\cal S} d{\bf a} ⋅ ({\bf E} × {\bf B})</li>
|
||||
</ul>
|
||||
<p>
|
||||
\label{Gr(8.9)}
|
||||
@@ -1968,7 +1712,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
|
||||
<p>
|
||||
Energy per unit time, per unit area carried by EM fields:
|
||||
</p>
|
||||
<div class="core div" id="org5a8440f">
|
||||
<div class="core div" id="org31c447f">
|
||||
<p>
|
||||
{\bf Poynting vector}
|
||||
\[
|
||||
@@ -1981,7 +1725,7 @@ Energy per unit time, per unit area carried by EM fields:
|
||||
<p>
|
||||
We can thus express Poynting's theorem more compactly:
|
||||
</p>
|
||||
<div class="core div" id="org70a18d5">
|
||||
<div class="core div" id="org0a1de2b">
|
||||
<p>
|
||||
{\bf Poynting's theorem}
|
||||
\[
|
||||
@@ -1994,7 +1738,7 @@ We can thus express Poynting's theorem more compactly:
|
||||
<p>
|
||||
where we have defined the total
|
||||
</p>
|
||||
<div class="core div" id="orgbd2e763">
|
||||
<div class="core div" id="org18b8949">
|
||||
<p>
|
||||
{\bf Energy in electromagnetic fields}
|
||||
\[
|
||||
@@ -2017,7 +1761,7 @@ Then,
|
||||
\]
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="org785fb4f">
|
||||
<div class="core div" id="org3f5d20d">
|
||||
<p>
|
||||
{\bf Poynting theorem (differential form)}
|
||||
\[
|
||||
@@ -2034,7 +1778,7 @@ and has a similar for to the continuity equation
|
||||
|
||||
|
||||
|
||||
<div class="example div" id="org542e42c">
|
||||
<div class="example div" id="org9d76705">
|
||||
<p>
|
||||
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
|
||||
Assuming that the field is uniform, the electric field parallel to the wire is
|
||||
@@ -2066,7 +1810,7 @@ and the value is as expected.
|
||||
|
||||
<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