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- <a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>
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- <details>
- <summary>
- Table of contents
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./in.html#in">Introduction</a><span class="headline-id">in</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./in_p.html#in_p">Preface</a><span class="headline-id">in.p</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./in_t.html#in_t">Tips for the reader</a><span class="headline-id">in.t</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
-
- </li>
- <li>
- <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems.html#ems">Electromagnetostatics</a><span class="headline-id">ems</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es.html#ems_es">Electrostatics</a><span class="headline-id">ems.es</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ec.html#ems_es_ec">Electric Charge</a><span class="headline-id">ems.es.ec</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ec_b.html#ems_es_ec_b">Basics</a><span class="headline-id">ems.es.ec.b</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_c.html#ems_es_ec_c">Conservation</a><span class="headline-id">ems.es.ec.c</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_q.html#ems_es_ec_q">Quantization</a><span class="headline-id">ems.es.ec.q</span>
-
- </li>
- <li>
- <a href="./ems_es_ec_s.html#ems_es_ec_s">Structure</a><span class="headline-id">ems.es.ec.s</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_efo.html#ems_es_efo">Electric Force and Energy</a><span class="headline-id">ems.es.efo</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_efo_cl.html#ems_es_efo_cl">Coulomb's Law</a><span class="headline-id">ems.es.efo.cl</span>
-
- </li>
- <li>
- <a href="./ems_es_efo_ps.html#ems_es_efo_ps">Principle of Superposition</a><span class="headline-id">ems.es.efo.ps</span>
-
- </li>
- <li>
- <a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>
-
- </li>
- <li>
- <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>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ef.html#ems_es_ef">Electrostatic Fields</a><span class="headline-id">ems.es.ef</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ef_pc.html#ems_es_ef_pc">Electrostatic Field of Point Charges</a><span class="headline-id">ems.es.ef.pc</span>
-
- </li>
- <li>
- <a href="./ems_es_ef_ccd.html#ems_es_ef_ccd">Electrostatic Field of Continuous Charge Distributions</a><span class="headline-id">ems.es.ef.ccd</span>
-
- </li>
- <li>
- <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>
-
- </li>
- <li>
- <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>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_ep_d.html#ems_es_ep_d">Definition</a><span class="headline-id">ems.es.ep.d</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
-
- </li>
- <li>
- <a href="./ems_es_ep_bc.html#ems_es_ep_bc">Electrostatic Boundary Conditions</a><span class="headline-id">ems.es.ep.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_es_c.html#ems_es_c">Conductors</a><span class="headline-id">ems.es.c</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_es_c_p.html#ems_es_c_p">Properties</a><span class="headline-id">ems.es.c.p</span>
-
- </li>
- <li>
- <a href="./ems_es_c_ic.html#ems_es_c_ic">Induced Charges</a><span class="headline-id">ems.es.c.ic</span>
-
- </li>
- <li>
- <a href="./ems_es_c_sc.html#ems_es_c_sc">Surface Charge and the Force on a Conductor</a><span class="headline-id">ems.es.c.sc</span>
-
- </li>
- <li>
- <a href="./ems_es_c_cap.html#ems_es_c_cap">Capacitors</a><span class="headline-id">ems.es.c.cap</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca.html#ems_ca">Calculating or Approximating the Electrostatic Potential</a><span class="headline-id">ems.ca</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_fe.html#ems_ca_fe">Fundamental Equations for the Electrostatic Potential</a><span class="headline-id">ems.ca.fe</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>
-
- </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>
-
- </li>
- <li>
- <a href="./ems_ca_fe_uP.html#ems_ca_fe_uP">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_mi.html#ems_ca_mi">The Method of Images</a><span class="headline-id">ems.ca.mi</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_mi_isc.html#ems_ca_mi_isc">Induced Surface Charges</a><span class="headline-id">ems.ca.mi.isc</span>
-
- </li>
- <li>
- <a href="./ems_ca_mi_fe.html#ems_ca_mi_fe">Force and Energy</a><span class="headline-id">ems.ca.mi.fe</span>
-
- </li>
- <li>
- <a href="./ems_ca_mi_o.html#ems_ca_mi_o">Other Image Problems</a><span class="headline-id">ems.ca.mi.o</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_sv.html#ems_ca_sv">Separation of Variables</a><span class="headline-id">ems.ca.sv</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_sv_car.html#ems_ca_sv_car">Cartesian Coordinates</a><span class="headline-id">ems.ca.sv.car</span>
-
- </li>
- <li>
- <a href="./ems_ca_sv_cyl.html#ems_ca_sv_cyl">Cylindrical Coordinates</a><span class="headline-id">ems.ca.sv.cyl</span>
-
- </li>
- <li>
- <a href="./ems_ca_sv_sph.html#ems_ca_sv_sph">Spherical Coordinates</a><span class="headline-id">ems.ca.sv.sph</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ca_me.html#ems_ca_me">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ca_me_a.html#ems_ca_me_a">Approximate Potential at Large Distance</a><span class="headline-id">ems.ca.me.a</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_md.html#ems_ca_me_md">Monopole and Dipole Terms</a><span class="headline-id">ems.ca.me.md</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_h.html#ems_ca_me_h">Higher Moments</a><span class="headline-id">ems.ca.me.h</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_Ed.html#ems_ca_me_Ed">The Electric Field of a Dipole</a><span class="headline-id">ems.ca.me.Ed</span>
-
- </li>
- <li>
- <a href="./ems_ca_me_Eq.html#ems_ca_me_Eq">The Electric Field of a Quadrupole</a><span class="headline-id">ems.ca.me.Eq</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms.html#ems_ms">Magnetostatics</a><span class="headline-id">ems.ms</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_lf.html#ems_ms_lf">Charges in Motion: the Lorentz Force Law</a><span class="headline-id">ems.ms.lf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
-
- </li>
- <li>
- <a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
-
- </li>
- <li>
- <a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_dcB.html#ems_ms_dcB">Divergence and Curl of \({\bf B}\)</a><span class="headline-id">ems.ms.dcB</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
-
- </li>
- <li>
- <a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
-
- </li>
- <li>
- <a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./ems_ms_vp.html#ems_ms_vp">The Vector Potential</a><span class="headline-id">ems.ms.vp</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_me.html#ems_ms_vp_me">Multipole Expansion of the Vector Potential</a><span class="headline-id">ems.ms.vp.me</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_comp.html#ems_ms_vp_comp">Comparison of Electrostatics and Magnetostatics</a><span class="headline-id">ems.ms.vp.comp</span>
-
- </li>
- <li>
- <a href="./ems_ms_vp_LC.html#ems_ms_vp_LC">The Levi-Civita Symbol</a><span class="headline-id">ems.ms.vp.LC</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details open="">
- <summary class="toc-open">
- <a href="./emsm.html#emsm">Electromagnetostatics in matter</a><span class="headline-id">emsm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details open="">
- <summary class="toc-open">
- <a href="./emsm_esm.html#emsm_esm">Electrostatics in matter</a><span class="headline-id">emsm.esm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>
-
- </li>
- <li>
- <a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details open="">
- <summary class="toc-currentpage">
- <a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
-
- </li>
- <li>
- <a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_D.html#emsm_esm_D">The Electric Displacement</a><span class="headline-id">emsm.esm.D</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_D_bc.html#emsm_esm_D_bc">Boundary Conditions</a><span class="headline-id">emsm.esm.D.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
- <a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>
-
- </li>
- <li>
- <a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm.html#emsm_msm">Magnetostatics in matter</a><span class="headline-id">emsm.msm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_m.html#emsm_msm_m">Magnetization</a><span class="headline-id">emsm.msm.m</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>
-
- </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>
-
- </li>
- <li>
- <a href="./emsm_msm_a.html#emsm_msm_a">Effect of Magnetic Field on Atomic Orbits</a><span class="headline-id">emsm.msm.a</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_fmo.html#emsm_msm_fmo">The Field of a Magnetized Object</a><span class="headline-id">emsm.msm.fmo</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_fmo_bc.html#emsm_msm_fmo_bc">Bound Currents</a><span class="headline-id">emsm.msm.fmo.bc</span>
-
- </li>
- <li>
- <a href="./emsm_msm_fmo_pibc.html#emsm_msm_fmo_pibc">Physical Interpretation of Bound Currents</a><span class="headline-id">emsm.msm.fmo.pibc</span>
-
- </li>
- <li>
- <a href="./emsm_msm_fmo_fim.html#emsm_msm_fmo_fim">The Magnetic Field Inside Matter</a><span class="headline-id">emsm.msm.fmo.fim</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_H.html#emsm_msm_H">The H Field</a><span class="headline-id">emsm.msm.H</span>
-
-
- </summary>
- <ul>
- <li>
- <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>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emsm_msm_lnlm.html#emsm_msm_lnlm">Linear and Nonlinear Media</a><span class="headline-id">emsm.msm.lnlm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emsm_msm_lnlm_sp.html#emsm_msm_lnlm_sp">Magnetic Susceptibility and Permeability</a><span class="headline-id">emsm.msm.lnlm.sp</span>
-
- </li>
- <li>
- <a href="./emsm_msm_lnlm_fm.html#emsm_msm_lnlm_fm">Ferromagnetism</a><span class="headline-id">emsm.msm.lnlm.fm</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd.html#emd">Electromagnetodynamics</a><span class="headline-id">emd</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emd_Fl.html#emd_Fl">Induction: Faraday's Law</a><span class="headline-id">emd.Fl</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_Fl_Fl.html#emd_Fl_Fl">Faraday's Law</a><span class="headline-id">emd.Fl.Fl</span>
-
- </li>
- <li>
- <a href="./emd_Fl_ief.html#emd_Fl_ief">The Induced Electric Field</a><span class="headline-id">emd.Fl.ief</span>
-
- </li>
- <li>
- <a href="./emd_Fl_i.html#emd_Fl_i">Inductance</a><span class="headline-id">emd.Fl.i</span>
-
- </li>
- <li>
- <a href="./emd_Fl_e.html#emd_Fl_e">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_Me.html#emd_Me">Maxwell's Equations</a><span class="headline-id">emd.Me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_Me_ebM.html#emd_Me_ebM">Electrodynamics Before Maxwell</a><span class="headline-id">emd.Me.ebM</span>
-
- </li>
- <li>
- <a href="./emd_Me_dc.html#emd_Me_dc">Maxwell's Correction to Ampère's Law; the Displacement Current</a><span class="headline-id">emd.Me.dc</span>
-
- </li>
- <li>
- <a href="./emd_Me_Me.html#emd_Me_Me">Maxwell's Equations</a><span class="headline-id">emd.Me.Me</span>
-
- </li>
- <li>
- <a href="./emd_Me_mc.html#emd_Me_mc">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_ce.html#emd_ce">Charge and Energy Flows</a><span class="headline-id">emd.ce</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_ce_ce.html#emd_ce_ce">The Continuity Equation</a><span class="headline-id">emd.ce.ce</span>
-
- </li>
- <li>
- <a href="./emd_ce_poy.html#emd_ce_poy">Poynting's Theorem; the Poynting Vector</a><span class="headline-id">emd.ce.poy</span>
-
- </li>
- <li>
- <a href="./emd_ce_mst.html#emd_ce_mst">Maxwell's Stress Tensor</a><span class="headline-id">emd.ce.mst</span>
-
- </li>
- <li>
- <a href="./emd_ce_mom.html#emd_ce_mom">Momentum</a><span class="headline-id">emd.ce.mom</span>
-
- </li>
- <li>
- <a href="./emd_ce_amom.html#emd_ce_amom">Angular Momentum</a><span class="headline-id">emd.ce.amom</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emd_emw.html#emd_emw">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emd_emw_we.html#emd_emw_we">The Wave Equation</a><span class="headline-id">emd.emw.we</span>
-
- </li>
- <li>
- <a href="./emd_emw_mpw.html#emd_emw_mpw">Monochromatic Plane Waves</a><span class="headline-id">emd.emw.mpw</span>
-
- </li>
- <li>
- <a href="./emd_emw_ep.html#emd_emw_ep">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm.html#emdm">Electromagnetodynamics in Matter</a><span class="headline-id">emdm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_Me.html#emdm_Me">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_Me_Mem.html#emdm_Me_Mem">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me.Mem</span>
-
- </li>
- <li>
- <a href="./emdm_Me_bc.html#emdm_Me_bc">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm.html#emdm_emwm">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_plm.html#emdm_emwm_plm">Propagation in Linear Media</a><span class="headline-id">emdm.emwm.plm</span>
-
- </li>
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- <a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>
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- <summary>
- <a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>
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- <ul>
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- <a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>
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- <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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- <ul>
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- <a href="./emdm_emwm_ad_c.html#emdm_emwm_ad_c">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span>
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- <a href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw">Guided waves</a><span class="headline-id">emdm.emwm.wg.gw</span>
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- <a href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r">Rectangular Waveguides</a><span class="headline-id">emdm.emwm.wg.r</span>
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- <a href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span>
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- <ul>
- <li>
- <a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>
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- <summary>
- <a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>
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- <ul>
- <li>
- <a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>
-
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- <li>
- <a href="./emf_g_Lg.html#emf_g_Lg">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span>
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- <ul>
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- <a href="./red_sr_p.html#red_sr_p">Postulates and their consequences</a><span class="headline-id">red.sr.p</span>
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- </li>
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- <a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>
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- </li>
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- <a href="./red_sr_4v.html#red_sr_4v">Covariant and Contravariant Four-Vectors</a><span class="headline-id">red.sr.4v</span>
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- <ul>
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- <a href="./red_rm_pt.html#red_rm_pt">Proper Time and Proper Velocity</a><span class="headline-id">red.rm.pt</span>
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- <a href="./red_rm_rme.html#red_rm_rme">Relativistic Momentum and Energy</a><span class="headline-id">red.rm.rme</span>
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- <a href="./red_rm_Mf.html#red_rm_Mf">Relativistic version of Newton's Laws; the Minkowski Force</a><span class="headline-id">red.rm.Mf</span>
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- <ul>
- <li>
- <a href="./red_rem_mre.html#red_rem_mre">Magnetism as a Relativistic Effect</a><span class="headline-id">red.rem.mre</span>
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- <a href="./red_rem_Ltf.html#red_rem_Ltf">Lorentz Transformation of Electromagnetic Fields</a><span class="headline-id">red.rem.Ltf</span>
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- <a href="./red_rem_Fmunu.html#red_rem_Fmunu">The Field Tensor</a><span class="headline-id">red.rem.Fmunu</span>
-
- </li>
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- <a href="./red_rem_Me.html#red_rem_Me">Maxwell's Equations in Relativistic Notation</a><span class="headline-id">red.rem.Me</span>
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- <a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>
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-
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- <ul>
- <li>
- <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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- <a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>
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- <a href="./d_emsm_msm.html#d_emsm_msm">Diagnostics: Magnetostatics in Matter</a><span class="headline-id">d.emsm.msm</span>
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- <a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>
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- <a href="./c_m_dc_del.html#c_m_dc_del">The \({\boldsymbol \nabla}\) Operator</a><span class="headline-id">c.m.dc.del</span>
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- <a href="./c_m_dc_div.html#c_m_dc_div">The Divergence</a><span class="headline-id">c.m.dc.div</span>
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- <a href="./c_m_dc_curl.html#c_m_dc_curl">The Curl</a><span class="headline-id">c.m.dc.curl</span>
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- <li>
- <a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
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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>
- <a href="./c_m_ic.html#c_m_ic">Integral Calculus</a><span class="headline-id">c.m.ic</span>
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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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- <a href="./c_m_ic_ftg.html#c_m_ic_ftg">The Fundamental Theorem for Gradients</a><span class="headline-id">c.m.ic.ftg</span>
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- <a href="./c_m_ic_gauss.html#c_m_ic_gauss">Gauss' Theorem</a><span class="headline-id">c.m.ic.gauss</span>
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- <a href="./c_m_ic_stokes.html#c_m_ic_stokes">Stokes' Theorem</a><span class="headline-id">c.m.ic.stokes</span>
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- <a href="./c_m_ic_ip.html#c_m_ic_ip">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span>
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- <a href="./c_m_dd_1d.html#c_m_dd_1d">The One-Dimensional Dirac Delta Function</a><span class="headline-id">c.m.dd.1d</span>
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- <h4 id="emsm_esm_po">Polarized Objects; Bound Charges<a class="headline-permalink" href="./emsm_esm_po.html#emsm_esm_po"><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><span class="headline-id">emsm.esm.po</span></h4>
-
- <div class="outline-text-4" id="text-emsm_esm_po">
- <p>
- What is the electric field produced by an object with polarization \({\bf P}\)?
- Let us try to address this question by Working with the potential.
- For a single dipole, we have <a href="./ems_ca_me_md.html#p_di">p_di</a>
- \[
- \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{({\bf r} - {\bf r}') \cdot {\bf p}}{|{\bf r} - {\bf r}'|^3}
- \label{Gr(4.8)}
- \]
- Considering a dipole moment per unit volume \({\bf P}\) (our definition of the polarization), we get
- </p>
- <div class="eqlabel" id="orgf9d8b73">
- <p>
- <a id="p_P"></a><a href="./emsm_esm_po.html#p_P"><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>
- </p>
- <div class="alteqlabels" id="org0d6708a">
- <ul class="org-ul">
- <li>Gr (4.9)</li>
- </ul>
-
- </div>
-
- </div>
- <p>
- \[
- \phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\cal V} d\tau' \frac{({\bf r} - {\bf r}') \cdot {\bf P}({\bf r}')}{|{\bf r} - {\bf r}'|^3}
- \tag{p_P}\label{p_P}
- \]
- This equation is perfectly acceptable and usable as it is.
- There exists however another convenient representation. Starting from
- \[
- {\boldsymbol \nabla}' \frac{1}{|{\bf r} - {\bf r}'|} = \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
- \]
- we can write (using product rule <a href="./c_m_dc_pr.html#div_prod">div_prod</a>)
- </p>
- \begin{align*}
- \phi({\bf r}) &= \frac{1}{4\pi\varepsilon_0} \int_{\cal V} d\tau' {\bf P} ({\bf r}') \cdot
- {\boldsymbol \nabla}' \frac{1}{|{\bf r} - {\bf r}'|} \nonumber \\
- & = \frac{1}{4\pi \varepsilon_0} \left[ \int_{\cal V} d\tau' {\boldsymbol \nabla}' \cdot \left( \frac{{\bf P} ({\bf r}')}{|{\bf r} - {\bf r}'|} \right)
- - \int_{\cal V} d\tau' \frac{1}{|{\bf r} - {\bf r}'|} {\boldsymbol \nabla}' \cdot {\bf P} ({\bf r}') \right]
- \end{align*}
- <p>
- Using the divergence theorem, this becomes
- </p>
- \begin{equation*}
- \phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \oint_{\cal S} d{\bf a}' \cdot \frac{{\bf P} ({\bf r}')}{|{\bf r} - {\bf r}'|} - \frac{1}{4\pi\varepsilon_0} \int_{\cal V} d\tau' \frac{{\boldsymbol \nabla}' \cdot {\bf P}({\bf r}')}{|{\bf r} - {\bf r}'|}.
- \end{equation*}
- <p>
- Interpretation: first terms is like contribution of a surface charge,
- </p>
- <div class="main div" id="orgda7ad07">
- <div class="eqlabel" id="orgb0c539d">
- <p>
- <a id="sigmab"></a><a href="./emsm_esm_po.html#sigmab"><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>
- </p>
- <div class="alteqlabels" id="org470a253">
- <ul class="org-ul">
- <li>Gr (4.11)</li>
- </ul>
-
- </div>
-
- </div>
- <p>
- \[
- \sigma_b({\bf r}) = {\bf P} ({\bf r}) \cdot \hat{\bf n}
- \tag{sigmab}\label{sigmab}
- \]
- </p>
-
- </div>
- <p>
- and second term looks like contribution of a volume charge,
- </p>
- <div class="main div" id="org2fcf5d5">
- <div class="eqlabel" id="org236705d">
- <p>
- <a id="rhob"></a><a href="./emsm_esm_po.html#rhob"><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>
- </p>
- <div class="alteqlabels" id="orgfa342f9">
- <ul class="org-ul">
- <li>Gr (4.12)</li>
- </ul>
-
- </div>
-
- </div>
- <p>
- \[
- \rho_b ({\bf r}) = -{\boldsymbol \nabla} \cdot {\bf P} ({\bf r})
- \tag{rhob}\label{rhob}
- \]
- </p>
-
- </div>
- <p>
- Using these definitions,
- </p>
- <div class="main div" id="org52f84a1">
- <div class="eqlabel" id="org6d08561">
- <p>
- <a id="p_bound"></a><a href="./emsm_esm_po.html#p_bound"><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>
- </p>
- <div class="alteqlabels" id="orgb8910a7">
- <ul class="org-ul">
- <li>Gr (4.13)</li>
- </ul>
-
- </div>
-
- </div>
- <p>
- \[
- \phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \oint_{\cal S} da' \frac{\sigma_b ({\bf r}')}{|{\bf r} - {\bf r}'|} + \frac{1}{4\pi \varepsilon_0} \int_{\cal V} d\tau' \frac{\rho_b ({\bf r}')}{|{\bf r} - {\bf r}'|}.
- \tag{p_bound}\label{p_bound}
- \]
- </p>
-
- </div>
- <p>
- These <b>bound charges</b> faithfully represent the object's sources for electrical fields.
- </p>
-
-
- <div class="example div" id="orge4443d6">
- <p>
- <b>Example: uniformly polarized sphere</b>
- </p>
-
- <p>
- <b>Task</b>: compute the electric field produced by a uniformly polarized sphere of radius \(R\).
- </p>
-
- <p>
- <b>Solution</b>: put \(z\) axis along \({\bf P}\). Since \({\bf P}\) is uniform, \(\rho_b = 0\).
- </p>
-
- <p>
- Surface charge:
- \[
- \sigma_b ({\bf r}) = {\bf P} \cdot \hat{\bf n} = P \cos \theta.
- \]
- This was computed in Example: surface charge density on a sphere (eq. <a href="./ems_ca_sv_sph.html#p_uni_ch_sph">p_uni_ch_sph</a>):
- \[
- \phi(r, \theta) = \left\{ \begin{array}{cc}
- \frac{P}{3\varepsilon_0} r\cos \theta, & r \leq R \\
- \frac{P}{3\varepsilon_0} \frac{R^3}{r^2} \cos \theta, & r \geq R.
- \end{array} \right.
- \]
- </p>
-
- <p>
- But \(r\cos \theta = z\), so the field inside the sphere is uniform,
- </p>
-
- <p>
- \[
- {\bf E} = -{\boldsymbol \nabla} \phi = -\frac{P}{3\varepsilon_0} \hat{\bf z} = -\frac{1}{3\varepsilon_0} {\bf P},
- \hspace{1cm} r < R.
- \]
- Outside the sphere, the potential is identical to that of pure point dipole at origin,
- </p>
-
- <p>
- \[
- \phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \frac{{\bf p} \cdot {\hat {\bf r}}}{r^2},
- \hspace{1cm} r > R
- \]
- where the total dipole moment is simply the integral over the polarization,
- </p>
-
- <p>
- \[
- {\bf p} = \frac{4}{3} \pi R^3 {\bf P}
- \]
- </p>
-
- </div>
- </div>
-
-
- <h5>In this section:</h5>
- <ul class="child-links-list">
- <li><a href="emsm_esm_po_pibc.html">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span></li>
- <li><a href="emsm_esm_po_fid.html">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span></li>
- </ul>
-
- <br><ul class="navigation-links"><li>Prev: <a href="emsm_esm_mE_P.html">Polarization <small>[emsm.esm.mE.P]</small></a></li><li>Next: <a href="emsm_esm_po_pibc.html">Physical Interpretation of Bound Charges <small>[emsm.esm.po.pibc]</small></a></li><li>Up: <a href="emsm_esm.html">Electrostatics in matter <small>[emsm.esm]</small></a></li></ul>
- <br>
- <hr>
- <div class="license">
- <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
- target="_blank" class="m-2">
- <img alt="Creative Commons License" style="border-width:0"
- src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
- </a>
- Except where otherwise noted, all content is licensed under a
- <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
- target="_blank">Creative Commons Attribution 4.0 International License</a>.
- </div>
- <div id="postamble" class="status">
- <p class="author">Author: Jean-Sébastien Caux</p>
- <p class="date">Created: 2022-03-24 Thu 08:42</p>
- <p class="validation"></p>
- </div>
-
-
- </div>
- </html>
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