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<div id="content">
<header>
<h1 class="title">
<a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>
</h1>
</header>
<nav id="collapsed-table-of-contents">
<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 open="">
<summary class="toc-open">
<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 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>

</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>
<li>
<a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>

</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>
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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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</li>

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<a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>


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<a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>

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<a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>


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<a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>

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<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>

</li>

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</li>

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<a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>


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<a href="./red_sr.html#red_sr">Special Relativity</a><span class="headline-id">red.sr</span>


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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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<a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>

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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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<a href="./red_rm.html#red_rm">Relativistic Mechanics</a><span class="headline-id">red.rm</span>


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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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<a href="./red_rem.html#red_rem">Relativistic Electromagnetism</a><span class="headline-id">red.rem</span>


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<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>

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<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>

</li>

</ul>
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<details>
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<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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<a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>


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<li>
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>

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<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>

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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_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</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="./d_emd_ce.html#d_emd_ce">Diagnostics: Conservation Laws</a><span class="headline-id">d.emd.ce</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>

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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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<a href="./a.html#a">Appendices</a><span class="headline-id">a</span>


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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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<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_sp.html#c_m_va_sp">Scalar product</a><span class="headline-id">c.m.va.sp</span>

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<a href="./c_m_va_cp.html#c_m_va_cp">Cross product</a><span class="headline-id">c.m.va.cp</span>

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<a href="./c_m_va_tp.html#c_m_va_tp">Triple Products</a><span class="headline-id">c.m.va.tp</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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<a href="./c_m_dc.html#c_m_dc">Differential Calculus</a><span class="headline-id">c.m.dc</span>


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<li>
<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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<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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<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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<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_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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<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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<li>
<a href="./c_m_ic_gauss.html#c_m_ic_gauss">Gauss' Theorem</a><span class="headline-id">c.m.ic.gauss</span>

</li>
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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>

</li>
<li>
<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_cs.html#c_m_cs">Coordinate Systems</a><span class="headline-id">c.m.cs</span>


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<a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>

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<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_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>

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<a href="./c_m_dd.html#c_m_dd">Dirac delta Distribution</a><span class="headline-id">c.m.dd</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>

</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.html#c_m_vf">Vector Fields</a><span class="headline-id">c.m.vf</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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<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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<a href="./c_m_uf_sph.html#c_m_uf_sph">Spherical coordinates</a><span class="headline-id">c.m.uf.sph</span>

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<a href="./c_m_uf_vi.html#c_m_uf_vi">Vector identities</a><span class="headline-id">c.m.uf.vi</span>

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<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ca.html">Calculating or Approximating the Electrostatic Potential</a></li><li><a class="breadcrumb-link"href="ems_ca_sv.html">Separation of Variables</a></li><li>Spherical Coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_sv_cyl.html">Cylindrical Coordinates&emsp;<small>[ems.ca.sv.cyl]</small></a></li><li>Next:&nbsp;<a href="ems_ca_me.html">The Multipole Expansion&emsp;<small>[ems.ca.me]</small></a></li><li>Up:&nbsp;<a href="ems_ca_sv.html">Separation of Variables&emsp;<small>[ems.ca.sv]</small></a></li></ul><div id="outline-container-ems_ca_sv_sph" class="outline-5">
<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">
  <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">ems.ca.sv.sph</span></h5>
<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) &amp;= 1 \nonumber \\
P_1 (x) &amp;= x \nonumber \\
P_2 (x) &amp;= \frac{1}{2} (3x^2 - 1) \nonumber \\
P_3 (x) &amp;= \frac{1}{2} (5x^3 - 3x) \nonumber \\
P_4 (x) &amp;= \frac{1}{8} (35x^4 - 30x^2 + 3) \nonumber \\
P_5 (x) &amp;= \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) &amp;= \frac{k}{3\varepsilon_0} r\cos \theta \hspace{3mm}\mbox{for}~ r \leq R,
\nonumber \\
\phi^o (r, \theta) &amp;= \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:&nbsp;<a href="ems_ca_sv_cyl.html">Cylindrical Coordinates&emsp;<small>[ems.ca.sv.cyl]</small></a></li><li>Next:&nbsp;<a href="ems_ca_me.html">The Multipole Expansion&emsp;<small>[ems.ca.me]</small></a></li><li>Up:&nbsp;<a href="ems_ca_sv.html">Separation of Variables&emsp;<small>[ems.ca.sv]</small></a></li></ul>
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</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>


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