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- Table of contents
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- <summary>
- <a href="./in.html#in">Introduction</a><span class="headline-id">in</span>
-
-
- </summary>
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- <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>
-
-
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- <li>
- <a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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- <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>
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- <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 open="">
- <summary class="toc-open">
- <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 open="">
- <summary class="toc-open">
- <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 class="toc-currentpage">
- <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>
- <summary>
- <a href="./emsm.html#emsm">Electromagnetostatics in matter</a><span class="headline-id">emsm</span>
-
-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <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>
- <summary>
- <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>
-
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- <a href="./emd_emw_ep.html#emd_emw_ep">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span>
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- </ul>
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- <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>
- <li>
- <a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>
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- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_ad.html#emdm_emwm_ad">Absorption and Dispersion</a><span class="headline-id">emdm.emwm.ad</span>
-
-
- </summary>
- <ul>
- <li>
- <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>
-
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emdm_emwm_wg.html#emdm_emwm_wg">Waveguides</a><span class="headline-id">emdm.emwm.wg</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw">Guided waves</a><span class="headline-id">emdm.emwm.wg.gw</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r">Rectangular Waveguides</a><span class="headline-id">emdm.emwm.wg.r</span>
-
- </li>
- <li>
- <a href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span>
-
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
-
- </ul>
- </details>
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>
-
- </li>
- <li>
-
- <details>
- <summary>
- <a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>
-
- </li>
- <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>
- </details>
- </li>
-
- </ul>
- </details>
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- <li>
-
- <details>
- <summary>
- <a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>
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-
- </summary>
- <ul>
- <li>
-
- <details>
- <summary>
- <a href="./red_sr.html#red_sr">Special Relativity</a><span class="headline-id">red.sr</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./red_sr_p.html#red_sr_p">Postulates and their consequences</a><span class="headline-id">red.sr.p</span>
-
- </li>
- <li>
- <a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>
-
- </li>
- <li>
- <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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- <details>
- <summary>
- <a href="./red_rm.html#red_rm">Relativistic Mechanics</a><span class="headline-id">red.rm</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./red_rm_pt.html#red_rm_pt">Proper Time and Proper Velocity</a><span class="headline-id">red.rm.pt</span>
-
- </li>
- <li>
- <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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- </li>
- <li>
- <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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-
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- </details>
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-
- <details>
- <summary>
- <a href="./red_rem.html#red_rem">Relativistic Electromagnetism</a><span class="headline-id">red.rem</span>
-
-
- </summary>
- <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>
-
- </li>
- <li>
- <a href="./red_rem_Ltf.html#red_rem_Ltf">Lorentz Transformation of Electromagnetic Fields</a><span class="headline-id">red.rem.Ltf</span>
-
- </li>
- <li>
- <a href="./red_rem_Fmunu.html#red_rem_Fmunu">The Field Tensor</a><span class="headline-id">red.rem.Fmunu</span>
-
- </li>
- <li>
- <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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-
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-
- </ul>
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- <details>
- <summary>
- <a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
-
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-
- </ul>
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- <details>
- <summary>
- <a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>
-
-
- </summary>
- <ul>
- <li>
- <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
-
- </li>
- <li>
- <a href="./d_ems_ca.html#d_ems_ca">Diagnostics: Calculating or Approximating the Electostatic Potential</a><span class="headline-id">d.ems.ca</span>
-
- </li>
- <li>
- <a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>
-
- </li>
- <li>
- <a href="./d_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</span>
-
- </li>
- <li>
- <a href="./d_emsm_msm.html#d_emsm_msm">Diagnostics: Magnetostatics in Matter</a><span class="headline-id">d.emsm.msm</span>
-
- </li>
- <li>
- <a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>
-
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- <a href="./d_emd_emw.html#d_emd_emw">Diagnostics: Electromagnetic Waves</a><span class="headline-id">d.emd.emw</span>
-
- </li>
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- <a href="./d_emf.html#d_emf">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span>
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- <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
-
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- <a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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- <summary>
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- <li>
- <a href="./a_l.html#a_l">Literature</a><span class="headline-id">a.l</span>
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- <a href="./c.html#c">Compendium</a><span class="headline-id">c</span>
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- <a href="./c_m.html#c_m">Mathematics</a><span class="headline-id">c.m</span>
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- <details>
- <summary>
- <a href="./c_m_va.html#c_m_va">Vector Analysis</a><span class="headline-id">c.m.va</span>
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- <a href="./c_m_va_n.html#c_m_va_n">Notation and algebraic properties</a><span class="headline-id">c.m.va.n</span>
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- <a href="./c_m_va_pds.html#c_m_va_pds">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span>
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- <details>
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- <a href="./c_m_dc.html#c_m_dc">Differential Calculus</a><span class="headline-id">c.m.dc</span>
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- <ul>
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- <a href="./c_m_dc_g.html#c_m_dc_g">Gradient</a><span class="headline-id">c.m.dc.g</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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- <li>
- <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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- <li>
- <a href="./c_m_dc_curl.html#c_m_dc_curl">The Curl</a><span class="headline-id">c.m.dc.curl</span>
-
- </li>
- <li>
- <a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
-
- </li>
- <li>
- <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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- <li>
-
- <details>
- <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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- </summary>
- <ul>
- <li>
- <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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- <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>
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- </li>
- <li>
- <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_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>
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- <li>
- <a href="./c_m_dd_div.html#c_m_dd_div">The Divergence of \(\hat{\bf r}/r^2\)</a><span class="headline-id">c.m.dd.div</span>
-
- </li>
- <li>
- <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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- </li>
- <li>
- <a href="./c_m_dd_3d.html#c_m_dd_3d">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span>
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- <a href="./c_m_vf_helm.html#c_m_vf_helm">The Helmholtz Theorem</a><span class="headline-id">c.m.vf.helm</span>
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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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- <details>
- <summary>
- <a href="./c_m_uf.html#c_m_uf">Useful Formulas</a><span class="headline-id">c.m.uf</span>
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- <a href="./c_m_uf_cyl.html#c_m_uf_cyl">Cylindrical coordinates</a><span class="headline-id">c.m.uf.cyl</span>
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- <h5 id="ems_ca_sv_sph">Spherical Coordinates<a class="headline-permalink" href="./ems_ca_sv_sph.html#ems_ca_sv_sph"><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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- <div class="outline-text-5" id="text-ems_ca_sv_sph">
- <p>
- In spherical coordinates, the Laplace equation takes the following form
- (using <a href="./c_m_cs_sph.html#sph_Lap">sph_Lap</a>):
- </p>
- <div class="eqlabel" id="org9243631">
- <p>
- <a id="Lap_sph"></a><a href="./ems_ca_sv_sph.html#Lap_sph"><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="orgc108761">
- <ul class="org-ul">
- <li>Gr (3.53)</li>
- <li>W (11-86)</li>
- </ul>
-
- </div>
-
- </div>
- <div class="main div" id="org2866001">
- <p>
-
- </p>
-
- \begin{equation}
- \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial \phi}{\partial r}\right)
- + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \phi}{\partial \theta}\right)
- + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \phi}{\partial \varphi^2} = 0
- \tag{Lap_sph}\label{Lap_sph}
- \end{equation}
-
- </div>
- <p>
- If you are dealing with a problem having <b>azimuthal symmetry</b>,
- \(\phi\) is independent of \(\varphi\) and the equation simplifies to:
- </p>
- <div class="eqlabel" id="org464b3a7">
- <p>
- <a id="Lap_sph_az"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az"><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="org9815bac">
- <ul class="org-ul">
- <li>Gr (3.54)</li>
- <li>W (11-87)</li>
- </ul>
-
- </div>
-
- </div>
-
- \begin{equation}
- \frac{\partial}{\partial r} \left(r^2 \frac{\partial \phi}{\partial r}\right)
- + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \phi}{\partial \theta}\right)
- = 0.
- \tag{Lap_sph_az}\label{Lap_sph_az}
- \end{equation}
-
- <p>
- Without loss of generality, we can look for a solution in the
- factorized product form
- </p>
-
- <p>
- \[
- \phi(r, \theta) = R(r) \Theta (\theta).
- \label{Gr(3.55)}
- \]
- </p>
-
- <p>
- Substituting this in <a href="./ems_ca_sv_sph.html#Lap_sph_az">Lap_sph_az</a> and dividing by \(\phi\) yields
- </p>
-
- <p>
- \[
- \frac{1}{R} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \frac{1}{\Theta \sin \theta} \frac{d}{d\theta}
- \left(\sin \theta \frac{d\Theta}{d\theta} \right) = 0.
- \label{Gr(3.56)}
- \]
- </p>
-
- <p>
- We can now apply the separation of variables logic: being dependent on
- separate variables, each term must be constant (the reasons for the
- convenient choice will become clear later),
- </p>
-
- <p>
- \[
- \frac{1}{R} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) = l(l+1), \hspace{1cm}
- \frac{1}{\Theta \sin \theta} \frac{d}{d\theta} \left(\sin \theta \frac{d\Theta}{d\theta} \right) = -l(l+1)
- \]
- </p>
-
- <p>
- We thus fall back onto ordinary differential equations, whereas our original
- problem involved partial differentials.
- </p>
-
- <p>
- Let us look at the radial equation first:
- </p>
-
- <p>
- \[
- \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) = l(l+1) R
- \]
- </p>
-
- <p>
- Let us search for a solution in the form \(r^\alpha\):
- since \(\frac{d}{dr} (r^2 \alpha r^{\alpha - 1}) = \alpha (\alpha + 1) r^{\alpha} = l(l+1) r^{\alpha}\), we get \(\alpha = l\) or \(-(l+1)\). The radial equation thus has the general solution
- </p>
-
- <p>
- \[
- R(r) = A r^l + \frac{B}{r^{l+1}}
- \]
- </p>
-
- <p>
- Separately from this, the angular equation takes the form
- </p>
-
- <p>
- \[
- \frac{d}{d\theta} \left(\sin \theta \frac{d\Theta}{d\theta} \right) = -l(l+1) \sin \theta ~\Theta
- \]
- </p>
-
- <p>
- This equation is solved by <b>Legendre polynomials</b> of the variable \(\cos \theta\):
- \[
- \Theta(\theta) = P_l (\cos \theta)
- \]
- </p>
-
-
- <div class="info div" id="orge95fc44">
- <p>
- <b>Legendre polynomials</b>
- </p>
-
- <p>
- When using spherical coordinates, one inevitably comes across integrals of the form
- \[
- \int_0^\pi d\theta ~\sin \theta ~f(\theta)
- \]
- for generic functions \(f\).
- </p>
-
- <p>
- Inspired by the logic of Fourier series, we would like to decompose such generic functions
- in a basis of "orthonormal" functions under this kind of integral (with the \(\sin \theta\) weight).
- This idea lead us to the <b>Legendre polynomials</b>, denoted \(P_l\), l = 0, 1, 2, …,
- and conveniently defined (for trigonometric arguments) to obey the orthogonality
- relationship (the reason for the normalization on the right-hand side will become clear later)
- </p>
-
- <div class="eqlabel" id="orgd196155">
- <p>
- <a id="Leg_orth_trig"></a><a href="./ems_ca_sv_sph.html#Leg_orth_trig"><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="org0daadc1">
-
- </div>
-
- </div>
- <p>
- \[
- \int_0^\pi d\theta \sin \theta ~P_l (\cos \theta) P_{l'} (\cos \theta) = \frac{2}{2l + 1} \delta_{l l'}
- \tag{Leg_orth_trig}\label{Leg_orth_trig}
- \]
- </p>
-
- <p>
- This same relation can be more simply written by using the variable \(x = \cos \theta\),
- </p>
- <div class="eqlabel" id="org760fd1b">
- <p>
- <a id="Leg_orth"></a><a href="./ems_ca_sv_sph.html#Leg_orth"><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="org23d0d7f">
-
- </div>
-
- </div>
-
- <p>
- \[
- \int_{-1}^1 dx P_l (x) P_{l'} (x) = \frac{2}{2l + 1} \delta_{l l'},
- \tag{Leg_orth}\label{Leg_orth}
- \]
- </p>
-
- <p>
- To get started, we need to define the "seed" polynomial (carrying label \(l=0\)).
- To make life easy, we set \(P_0 (x) = 1\). Higher polynomials are then sought in the
- form of power series in \(x\). This leads to the first few Legendre polynomials being:
- </p>
- <div class="eqlabel" id="orgdc31429">
- <p>
- <a id="Leg_pols"></a><a href="./ems_ca_sv_sph.html#Leg_pols"><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="org0f77fac">
-
- </div>
-
- </div>
-
- \begin{align}
- P_0 (x) &= 1 \nonumber \\
- P_1 (x) &= x \nonumber \\
- P_2 (x) &= \frac{1}{2} (3x^2 - 1) \nonumber \\
- P_3 (x) &= \frac{1}{2} (5x^3 - 3x) \nonumber \\
- P_4 (x) &= \frac{1}{8} (35x^4 - 30x^2 + 3) \nonumber \\
- P_5 (x) &= \frac{1}{8} (63x^5 - 70x^3 + 15x).
- \tag{Leg_pols}\label{Leg_pols}
- \end{align}
-
- <p>
- The prefactor (and thus the factor in the orthogonality relations <a href="./ems_ca_sv_sph.html#Leg_orth_trig">Leg_orth_trig</a> and the equivalent <a href="./ems_ca_sv_sph.html#Leg_orth">Leg_orth</a> is chosen for convenience such that each polynomial
- takes the value \(1\) when evaluated at argument \(x = 1\),
- </p>
- <div class="eqlabel" id="org4dfddbe">
- <p>
- <a id="Pl_1_1"></a><a href="./ems_ca_sv_sph.html#Pl_1_1"><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="org7633d23">
-
- </div>
-
- </div>
-
- <p>
- \[
- P_l(1) = 1
- \tag{Pl_1_1}\label{Pl_1_1}
- \]
- </p>
-
- <p>
- The Legendre polynomial \(P_l\) obeys the differential equation
- </p>
- <div class="eqlabel" id="org2b73241">
- <p>
- <a id="Leg_de_trig"></a><a href="./ems_ca_sv_sph.html#Leg_de_trig"><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="orgc06b5cd">
-
- </div>
-
- </div>
- <p>
- \[
- \left[\frac{d}{d\theta} \left( \sin \theta \frac{d}{d\theta} \right) + l (l+1) \sin \theta \right] P_l (\cos \theta) = 0.
- \tag{Leg_de_trig}\label{Leg_de_trig}
- \]
- or equivalently
- </p>
- <div class="eqlabel" id="orged631e3">
- <p>
- <a id="Leg_de"></a><a href="./ems_ca_sv_sph.html#Leg_de"><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="org3cb5fe1">
-
- </div>
-
- </div>
- <p>
- \[
- \left[(1 - x^2) \frac{d^2}{dx^2} - 2x \frac{d}{dx} + l(l+1) \right] P_l (x) = 0.
- \tag{Leg_de}\label{Leg_de}
- \]
- </p>
-
- <p>
- A particularly convenient formula for deriving \(P_l(x)\)
- is the <b>Rodrigues formula</b>:
- </p>
- <div class="eqlabel" id="org55ec248">
- <p>
- <a id="Rodrigues"></a><a href="./ems_ca_sv_sph.html#Rodrigues"><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="orgdee9d6b">
-
- </div>
-
- </div>
-
- <p>
- \[
- P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l
- \tag{Rodrigues}\label{Rodrigues}
- \]
- </p>
-
- <p>
- Actually, a more practical formula is <b>Bonnet's recursion relation</b>
- </p>
- <div class="eqlabel" id="org1d3acfd">
- <p>
- <a id="Bonnet"></a><a href="./ems_ca_sv_sph.html#Bonnet"><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="orgd631ecf">
-
- </div>
-
- </div>
-
- <p>
- \[
- (l + 1) P_{l+1} (x) = (2l + 1) x P_l (x) - l P_{l-1} (x)
- \tag{Bonnet}\label{Bonnet}
- \]
- </p>
-
- </div>
-
-
-
- <p>
- Going back to the angular equation, let us first remark that this
- is a second order equation, and should thus have
- 2 solutions. These other solutions blow up at \(\theta = 0\) and/or \(\theta = \pi\),
- and we thus exclude them on physical grounds. For example, the
- (here discarded) second solution for \(l = 0\) is
- </p>
-
- <p>
- \[
- \Theta (\theta) = \ln \left( \tan \frac{\theta}{2} \right)
- \]
- </p>
-
- <p>
- We therefore come to the culmination of our efforts here, and write
- the general solution to <i>any</i> problem with azimuthal symmetry
- (for which the potential takes a finite value for \(\theta = 0, \pi\)) as
- </p>
- <div class="eqlabel" id="orgb35f0cb">
- <p>
- <a id="Lap_sph_az_sol"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az_sol"><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="org8d2700b">
- <ul class="org-ul">
- <li>Gr (3.65)</li>
- </ul>
-
- </div>
-
- </div>
- <div class="main div" id="org8b57fc8">
- <p>
- \[
- \phi(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta)
- \tag{Lap_sph_az_sol}\label{Lap_sph_az_sol}
- \]
- </p>
-
- </div>
-
-
-
- <div class="example div" id="org03072a7">
- <p>
- <b>Example: potential inside a hollow sphere</b>
- </p>
-
- <p>
- Consider a hollow sphere of radius \(R\), with specified potential on the surface
- equal to a given function \(\phi_0 (\theta)\).
- </p>
-
- <p>
- <b>Question</b>: find potential inside the sphere.
- </p>
-
- <p>
- <b>Solution</b>: (by the way, this is a case of Dirichlet boundary conditions)
- </p>
-
- <p>
- Since the potential cannot diverge at the origin, we set \(B_l = 0\) \(\forall l\).
- </p>
-
- <p>
- Our solution must thus take the form
- </p>
-
- <p>
- \[
- \phi(r,\theta) = \sum_{l=0}^\infty A_l r^l P_l (\cos \theta)
- \label{Gr(3.66)}
- \]
- </p>
-
- <p>
- The specified boundary condition means that
- </p>
-
- <p>
- \[
- \phi(R,\theta) = \sum_{l=0}^\infty A_l R^l P_l (\cos \theta) = \phi_0 (\theta)
- \label{Gr(3.67)}
- \]
- </p>
-
- <p>
- We can now use the fact that Legendre polynomials are orthogonal functions, giving us
- </p>
-
- <p>
- \[
- A_l = \frac{2l + 1}{2R^l} \int_0^\pi d\theta \sin \theta ~P_l (\cos \theta) \phi_0 (\theta).
- \]
- </p>
-
- <p>
- <b>Specific example:</b> choose
- </p>
-
- <p>
- \[
- \phi_0 (\theta) = k \sin^2 (\theta/2)
- \label{Gr(3.70)}
- \]
- </p>
-
- <p>
- This is \(\phi_0 (\theta) = \frac{k}{2} (1 - \cos \theta) = \frac{k}{2} (P_0 (\cos \theta) - P_1 (\cos \theta))\).
- </p>
-
- <p>
- Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
- </p>
-
- <p>
- \[
- \phi(r, \theta) = \frac{k}{2} (1 - \frac{r}{R} \cos \theta).
- \label{Gr(3.71)}
- \]
- </p>
-
- </div>
-
-
- <div class="example div" id="orge6863ac">
- <p>
- <b>Example: surface charge density on sphere</b>
- </p>
-
- <p>
- Consider once again a spherical shell of radius \(R\).
- This time, we affix a surface charge density \(\sigma_0 (\theta)\)
- over the surface of the shell.
- </p>
-
- <p>
- <b>Question</b>: find \(\phi\) inside and outside sphere.
- </p>
-
- <p>
- <b>Solution</b>: (by the way, this is a case of Neumann boundary conditions)
- </p>
-
- <p>
- We could of course use direct integration of <a href="./ems_es_ep_d.html#p_scd">p_scd</a>, but let us save some
- effort by invoking separation of variables. In the interior of the shell,
- </p>
-
- <p>
- \[
- \phi^i (r,\theta) = \sum_{l=0}^{\infty} A_l^i r^l P_l (\cos \theta), \hspace{1cm} r \leq R
- \]
- </p>
-
- <p>
- (other terms blow up as \(r \rightarrow 0\), so we need to set \(B_l^i = 0\) here).
- </p>
-
- <p>
- In the region exterior to the shell,
- </p>
-
- <p>
- \[
- \phi^o(r, \theta) = \sum_{l=0}^{\infty} \frac{B_l^o}{r^{l+1}} P_l (\cos \theta), \hspace{1cm} r \geq R
- \]
- </p>
-
- <p>
- (other terms blow up as \(r \rightarrow \infty\), so we need to set \(A_l^o = 0\) here).
- </p>
-
- <p>
- Since the potential must be continuous at \(r = R\), we must have
- </p>
-
- <p>
- \[
- \sum_{l=0}^{\infty} A_l^i R^l P_l (\cos \theta) = \sum_{l=0}^{\infty} \frac{B_l^o}{R^{l+1}} P_l (\cos \theta)
- \]
- </p>
-
- <p>
- Invoking the orthononality of Legendre polynomials thus yields
- </p>
-
- <p>
- \[
- B_l^o = A_l^i R^{2l + 1}.
- \]
- </p>
-
- <p>
- The surface charge induces a discontinuity in derivative of \(\phi\)
- according to <a href="./ems_es_ep_bc.html#dpdisc">dpdisc</a>:
- </p>
-
- \begin{equation*}
- \left( \frac{\partial \phi^{o}}{\partial r} - \frac{\partial \phi^{i}}{\partial r} \right)
- = -\frac{\sigma_0 (\theta)}{\varepsilon_0}
- \end{equation*}
-
- <p>
- so
- </p>
-
- <p>
- \[
- -\sum_{l=0}^\infty \left((l+1) \frac{B_l^o}{R^{l+2}} + l A_l^i R^{l-1} \right) P_l (\cos \theta) = -\frac{\sigma_0 (\theta)}{\varepsilon_0},
- \]
- and thus
- </p>
-
- <p>
- \[
- \sum_{l=0}^\infty (2l+1) A_l^i R^{l-1} P_l (\cos \theta) = \frac{\sigma_0 (\theta)}{\varepsilon_0}
- \]
- </p>
-
- <p>
- The coefficients can be fixed from the orthogonality relation <a href="./ems_ca_sv_sph.html#Leg_orth_trig">Leg_orth_trig</a>,
- </p>
-
- <p>
- \[
- A_l^i = \frac{1}{2\varepsilon_0 R^{l-1}} \int_0^\pi d\theta \sin \theta ~\sigma_0 (\theta) P_l (\cos \theta).
- \]
- </p>
-
-
- <p>
- <b>Specific case</b>: choose
- </p>
-
- <p>
- \[
- \sigma_0 (\theta) = k \cos \theta = k P_1 (\cos \theta)
- \label{Gr(3.85)}
- \]
- </p>
-
- <p>
- All \(A_l^i = 0\) except for \(l = 1\), in which case
- </p>
-
- <p>
- \[
- A_1^i = \frac{k}{2\varepsilon_0} \int_0^\pi d\theta \sin \theta [P_l(\cos \theta)]^2 = \frac{k}{3\varepsilon_0}.
- \]
- </p>
-
- <p>
- The potential inside/outside the sphere is then
- </p>
- <div class="eqlabel" id="org18097a4">
- <p>
- <a id="p_uni_ch_sph"></a><a href="./ems_ca_sv_sph.html#p_uni_ch_sph"><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="orgce76611">
-
- </div>
-
- </div>
-
- \begin{align}
- \phi^i (r,\theta) &= \frac{k}{3\varepsilon_0} r\cos \theta \hspace{3mm}\mbox{for}~ r \leq R,
- \nonumber \\
- \phi^o (r, \theta) &= \frac{k R^3}{3\varepsilon_0} \frac{\cos \theta}{r^2} \hspace{3mm}\mbox{for}~ r \geq R.
- \tag{p_uni_ch_sph}\label{p_uni_ch_sph}
- \end{align}
-
- </div>
- </div>
- </div>
-
-
- <br><ul class="navigation-links"><li>Prev: <a href="ems_ca_sv_cyl.html">Cylindrical Coordinates <small>[ems.ca.sv.cyl]</small></a></li><li>Next: <a href="ems_ca_me.html">The Multipole Expansion <small>[ems.ca.me]</small></a></li><li>Up: <a href="ems_ca_sv.html">Separation of Variables <small>[ems.ca.sv]</small></a></li></ul>
- <br>
- <hr>
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- target="_blank" class="m-2">
- <img alt="Creative Commons License" style="border-width:0"
- src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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- 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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