2119 lines
56 KiB
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2119 lines
56 KiB
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<title>Pre-Quantum Electrodynamics</title>
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<h1 class="title">
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<a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>
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</h1>
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</header>
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<nav id="collapsed-table-of-contents">
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<details>
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<summary>
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Table of contents
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./in.html#in">Introduction</a><span class="headline-id">in</span>
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</summary>
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<ul>
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<li>
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<a href="./in_p.html#in_p">Preface</a><span class="headline-id">in.p</span>
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</li>
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<li>
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<details>
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<summary>
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<a href="./in_t.html#in_t">Tips for the reader</a><span class="headline-id">in.t</span>
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</summary>
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<ul>
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<li>
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<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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</li>
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<li>
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<a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details open="">
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<summary class="toc-open">
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<a href="./ems.html#ems">Electromagnetostatics</a><span class="headline-id">ems</span>
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./ems_es.html#ems_es">Electrostatics</a><span class="headline-id">ems.es</span>
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./ems_es_ec.html#ems_es_ec">Electric Charge</a><span class="headline-id">ems.es.ec</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_ec_b.html#ems_es_ec_b">Basics</a><span class="headline-id">ems.es.ec.b</span>
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</li>
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<li>
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<a href="./ems_es_ec_c.html#ems_es_ec_c">Conservation</a><span class="headline-id">ems.es.ec.c</span>
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</li>
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<li>
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<a href="./ems_es_ec_q.html#ems_es_ec_q">Quantization</a><span class="headline-id">ems.es.ec.q</span>
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</li>
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<li>
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<a href="./ems_es_ec_s.html#ems_es_ec_s">Structure</a><span class="headline-id">ems.es.ec.s</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_efo.html#ems_es_efo">Electric Force and Energy</a><span class="headline-id">ems.es.efo</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_efo_cl.html#ems_es_efo_cl">Coulomb's Law</a><span class="headline-id">ems.es.efo.cl</span>
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</li>
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<li>
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<a href="./ems_es_efo_ps.html#ems_es_efo_ps">Principle of Superposition</a><span class="headline-id">ems.es.efo.ps</span>
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</li>
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<li>
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<a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>
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</li>
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<li>
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<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>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_ef.html#ems_es_ef">Electrostatic Fields</a><span class="headline-id">ems.es.ef</span>
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</summary>
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<ul>
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<li>
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<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>
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</li>
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<li>
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<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>
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</li>
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<li>
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<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>
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</li>
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<li>
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<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>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_ep_d.html#ems_es_ep_d">Definition</a><span class="headline-id">ems.es.ep.d</span>
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</li>
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<li>
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<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>
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</li>
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<li>
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<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>
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</li>
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<li>
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<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>
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</li>
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<li>
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<a href="./ems_es_ep_bc.html#ems_es_ep_bc">Electrostatic Boundary Conditions</a><span class="headline-id">ems.es.ep.bc</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_c.html#ems_es_c">Conductors</a><span class="headline-id">ems.es.c</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_c_p.html#ems_es_c_p">Properties</a><span class="headline-id">ems.es.c.p</span>
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</li>
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<li>
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<a href="./ems_es_c_ic.html#ems_es_c_ic">Induced Charges</a><span class="headline-id">ems.es.c.ic</span>
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</li>
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<li>
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<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>
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</li>
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<li>
|
|
<a href="./ems_es_c_cap.html#ems_es_c_cap">Capacitors</a><span class="headline-id">ems.es.c.cap</span>
|
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</li>
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|
|
|
</ul>
|
|
</details>
|
|
</li>
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|
|
</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 open="">
|
|
<summary class="toc-open">
|
|
<a href="./ems_ca_fe.html#ems_ca_fe">Fundamental Equations for the Electrostatic Potential</a><span class="headline-id">ems.ca.fe</span>
|
|
|
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|
|
</summary>
|
|
<ul>
|
|
<li class="toc-currentpage">
|
|
<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>
|
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|
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</li>
|
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<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>
|
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|
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</li>
|
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|
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</ul>
|
|
</details>
|
|
</li>
|
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<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>
|
|
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|
|
</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>
|
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|
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</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>
|
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|
|
</ul>
|
|
</details>
|
|
</li>
|
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<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./ems_ca_sv.html#ems_ca_sv">Separation of Variables</a><span class="headline-id">ems.ca.sv</span>
|
|
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|
|
</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>
|
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|
|
</li>
|
|
<li>
|
|
<a href="./ems_ca_sv_cyl.html#ems_ca_sv_cyl">Cylindrical Coordinates</a><span class="headline-id">ems.ca.sv.cyl</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./ems_ca_sv_sph.html#ems_ca_sv_sph">Spherical Coordinates</a><span class="headline-id">ems.ca.sv.sph</span>
|
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|
|
</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>
|
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|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./ems_ms.html#ems_ms">Magnetostatics</a><span class="headline-id">ems.ms</span>
|
|
|
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|
|
</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>
|
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|
|
</ul>
|
|
</details>
|
|
</li>
|
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|
|
</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>
|
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|
|
</li>
|
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./emd_Me.html#emd_Me">Maxwell's Equations</a><span class="headline-id">emd.Me</span>
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</summary>
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<ul>
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<li>
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<a href="./emd_Me_ebM.html#emd_Me_ebM">Electrodynamics Before Maxwell</a><span class="headline-id">emd.Me.ebM</span>
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</li>
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<li>
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<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>
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</li>
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<li>
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<a href="./emd_Me_Me.html#emd_Me_Me">Maxwell's Equations</a><span class="headline-id">emd.Me.Me</span>
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</li>
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<li>
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<a href="./emd_Me_mc.html#emd_Me_mc">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./emd_ce.html#emd_ce">Charge and Energy Flows</a><span class="headline-id">emd.ce</span>
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</summary>
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<ul>
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<li>
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<a href="./emd_ce_ce.html#emd_ce_ce">The Continuity Equation</a><span class="headline-id">emd.ce.ce</span>
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</li>
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<li>
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<a href="./emd_ce_poy.html#emd_ce_poy">Poynting's Theorem; the Poynting Vector</a><span class="headline-id">emd.ce.poy</span>
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</li>
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<li>
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<a href="./emd_ce_mst.html#emd_ce_mst">Maxwell's Stress Tensor</a><span class="headline-id">emd.ce.mst</span>
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</li>
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<li>
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<a href="./emd_ce_mom.html#emd_ce_mom">Momentum</a><span class="headline-id">emd.ce.mom</span>
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</li>
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<li>
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<a href="./emd_ce_amom.html#emd_ce_amom">Angular Momentum</a><span class="headline-id">emd.ce.amom</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./emd_emw.html#emd_emw">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span>
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</summary>
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<ul>
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<li>
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<a href="./emd_emw_we.html#emd_emw_we">The Wave Equation</a><span class="headline-id">emd.emw.we</span>
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</li>
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<li>
|
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<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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</li>
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<li>
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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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</li>
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</ul>
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</details>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./emdm.html#emdm">Electromagnetodynamics in Matter</a><span class="headline-id">emdm</span>
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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<a href="./emdm_Me.html#emdm_Me">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me</span>
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</summary>
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<ul>
|
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<li>
|
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<a href="./emdm_Me_Mem.html#emdm_Me_Mem">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me.Mem</span>
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</li>
|
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<li>
|
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<a href="./emdm_Me_bc.html#emdm_Me_bc">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span>
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</li>
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</ul>
|
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./emdm_emwm.html#emdm_emwm">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span>
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</summary>
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<ul>
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<li>
|
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<a href="./emdm_emwm_plm.html#emdm_emwm_plm">Propagation in Linear Media</a><span class="headline-id">emdm.emwm.plm</span>
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</li>
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<li>
|
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<a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>
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</li>
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<li>
|
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<details>
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<summary>
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<a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>
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</summary>
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<ul>
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<li>
|
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<a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>
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</li>
|
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<li>
|
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<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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</li>
|
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<li>
|
|
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
|
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|
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</li>
|
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<li>
|
|
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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</li>
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</ul>
|
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</details>
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</li>
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<li>
|
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<details>
|
|
<summary>
|
|
<a href="./emdm_emwm_ad.html#emdm_emwm_ad">Absorption and Dispersion</a><span class="headline-id">emdm.emwm.ad</span>
|
|
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|
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</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>
|
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|
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</li>
|
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</ul>
|
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</details>
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</li>
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<li>
|
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|
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<details>
|
|
<summary>
|
|
<a href="./emdm_emwm_wg.html#emdm_emwm_wg">Waveguides</a><span class="headline-id">emdm.emwm.wg</span>
|
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|
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|
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</summary>
|
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<ul>
|
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<li>
|
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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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|
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</li>
|
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<li>
|
|
<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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|
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</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>
|
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</li>
|
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</ul>
|
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</details>
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</li>
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</ul>
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</details>
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</li>
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</ul>
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</details>
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</li>
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<li>
|
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|
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<details>
|
|
<summary>
|
|
<a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>
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|
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|
|
</summary>
|
|
<ul>
|
|
<li>
|
|
<a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>
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|
|
</li>
|
|
<li>
|
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|
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<details>
|
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<summary>
|
|
<a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>
|
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|
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</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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</li>
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</ul>
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</details>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
|
|
<summary>
|
|
<a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>
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|
|
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|
|
</summary>
|
|
<ul>
|
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<li>
|
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|
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<details>
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<summary>
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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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|
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|
|
</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>
|
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|
|
</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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</li>
|
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</ul>
|
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</details>
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</li>
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<li>
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<details>
|
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<summary>
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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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|
|
</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>
|
|
|
|
</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>
|
|
|
|
</li>
|
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|
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</ul>
|
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</details>
|
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</li>
|
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<li>
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<details>
|
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<summary>
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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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|
|
</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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</li>
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</ul>
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</details>
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</li>
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</ul>
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</details>
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</li>
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<li>
|
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<details>
|
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<summary>
|
|
<a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>
|
|
|
|
|
|
</summary>
|
|
<ul>
|
|
<li>
|
|
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
|
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|
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</li>
|
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</ul>
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</details>
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</li>
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<li>
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<details>
|
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<summary>
|
|
<a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>
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|
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</summary>
|
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<ul>
|
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<li>
|
|
<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
|
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|
|
</li>
|
|
<li>
|
|
<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
|
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|
|
</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>
|
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|
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</li>
|
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<li>
|
|
<a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>
|
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|
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</li>
|
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<li>
|
|
<a href="./d_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</span>
|
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|
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</li>
|
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<li>
|
|
<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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|
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</li>
|
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<li>
|
|
<a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>
|
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|
|
</li>
|
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<li>
|
|
<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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|
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</li>
|
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<li>
|
|
<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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</li>
|
|
<li>
|
|
<a href="./d_emf.html#d_emf">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span>
|
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</li>
|
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<li>
|
|
<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
|
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</li>
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</ul>
|
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</details>
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</li>
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<li>
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<details>
|
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<summary>
|
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<a href="./a.html#a">Appendices</a><span class="headline-id">a</span>
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|
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</summary>
|
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<ul>
|
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<li>
|
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<a href="./a_l.html#a_l">Literature</a><span class="headline-id">a.l</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
|
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<summary>
|
|
<a href="./c.html#c">Compendium</a><span class="headline-id">c</span>
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</summary>
|
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<ul>
|
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<li>
|
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<details>
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<summary>
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<a href="./c_m.html#c_m">Mathematics</a><span class="headline-id">c.m</span>
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</summary>
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<ul>
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<li>
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<details>
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<summary>
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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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</summary>
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<ul>
|
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<li>
|
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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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</li>
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<li>
|
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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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</li>
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<li>
|
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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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</li>
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<li>
|
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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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</li>
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<li>
|
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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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</li>
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</ul>
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</details>
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</li>
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<li>
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<details>
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<summary>
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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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</summary>
|
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<ul>
|
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<li>
|
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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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</li>
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<li>
|
|
<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>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_dc_div.html#c_m_dc_div">The Divergence</a><span class="headline-id">c.m.dc.div</span>
|
|
|
|
</li>
|
|
<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>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./c_m_ic.html#c_m_ic">Integral Calculus</a><span class="headline-id">c.m.ic</span>
|
|
|
|
|
|
</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>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_ic_ftc.html#c_m_ic_ftc">The Fundamental Theorem of Calculus</a><span class="headline-id">c.m.ic.ftc</span>
|
|
|
|
</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>
|
|
|
|
</li>
|
|
<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>
|
|
<li>
|
|
<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>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./c_m_cs.html#c_m_cs">Coordinate Systems</a><span class="headline-id">c.m.cs</span>
|
|
|
|
|
|
</summary>
|
|
<ul>
|
|
<li>
|
|
<a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./c_m_dd.html#c_m_dd">Dirac delta Distribution</a><span class="headline-id">c.m.dd</span>
|
|
|
|
|
|
</summary>
|
|
<ul>
|
|
<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>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./c_m_vf.html#c_m_vf">Vector Fields</a><span class="headline-id">c.m.vf</span>
|
|
|
|
|
|
</summary>
|
|
<ul>
|
|
<li>
|
|
<a href="./c_m_vf_helm.html#c_m_vf_helm">The Helmholtz Theorem</a><span class="headline-id">c.m.vf.helm</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
<li>
|
|
|
|
<details>
|
|
<summary>
|
|
<a href="./c_m_uf.html#c_m_uf">Useful Formulas</a><span class="headline-id">c.m.uf</span>
|
|
|
|
|
|
</summary>
|
|
<ul>
|
|
<li>
|
|
<a href="./c_m_uf_cyl.html#c_m_uf_cyl">Cylindrical coordinates</a><span class="headline-id">c.m.uf.cyl</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_uf_sph.html#c_m_uf_sph">Spherical coordinates</a><span class="headline-id">c.m.uf.sph</span>
|
|
|
|
</li>
|
|
<li>
|
|
<a href="./c_m_uf_vi.html#c_m_uf_vi">Vector identities</a><span class="headline-id">c.m.uf.vi</span>
|
|
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</li>
|
|
|
|
</ul>
|
|
</details>
|
|
</nav>
|
|
<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_fe.html">Fundamental Equations for the Electrostatic Potential</a></li><li>The Laplace Equation</li></ul><ul class="navigation-links"><li>Prev: <a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential <small>[ems.ca.fe]</small></a></li><li>Next: <a href="ems_ca_fe_g.html">Green's Identities <small>[ems.ca.fe.g]</small></a></li><li>Up: <a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential <small>[ems.ca.fe]</small></a></li></ul><div id="outline-container-ems_ca_fe_L" class="outline-5">
|
|
<h5 id="ems_ca_fe_L">The Laplace Equation<a class="headline-permalink" href="./ems_ca_fe_L.html#ems_ca_fe_L"><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.fe.L</span></h5>
|
|
<div class="outline-text-5" id="text-ems_ca_fe_L">
|
|
<p>
|
|
In regions of space where there is no charge density,
|
|
the potential must solve Laplace's equation.
|
|
Let us discuss how solutions to this equation look,
|
|
in increasingly complicated situations.
|
|
</p>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-ems_ca_fe_L_1d" class="outline-6">
|
|
<h6 id="ems_ca_fe_L_1d"><a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a></h6>
|
|
<div class="outline-text-6" id="text-ems_ca_fe_L_1d">
|
|
<p>
|
|
In one dimension, the potential is a single-variable
|
|
function \(\phi (x)\) and the Laplace equation reads
|
|
</p>
|
|
|
|
<div class="eqlabel" id="org46aafa7">
|
|
<p>
|
|
<a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><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="org459093f">
|
|
|
|
</div>
|
|
|
|
</div>
|
|
<p>
|
|
\[
|
|
\frac{d^2 \phi(x)}{dx^2} = 0.
|
|
\tag{Lap_1d}\label{Lap_1d}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
The solution to this is
|
|
</p>
|
|
<div class="eqlabel" id="orged9e79a">
|
|
<p>
|
|
<a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_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="org599dad5">
|
|
<ul class="org-ul">
|
|
<li>Gr (3.6)</li>
|
|
</ul>
|
|
|
|
</div>
|
|
|
|
</div>
|
|
<p>
|
|
\[
|
|
\phi(x) = a x + b
|
|
\tag{Lap_1d_sol}\label{Lap_1d_sol}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
Properties:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>
|
|
<b>Balance</b>: \(\phi(x)\) is the average of \(\phi(x + dx)\) and \(\phi(x - dx)\) for any \(dx\) (with \(x \pm dx\) still being in
|
|
the region where Laplace is satisfied, of course).</li>
|
|
<li>
|
|
<b>No extrema</b>: \(\phi(x)\) has no local extrema. Max/min
|
|
values must occur at boundaries.</li>
|
|
</ul>
|
|
|
|
|
|
<p>
|
|
In a particular problem, to fix the solution (said
|
|
otherwise: to fix the parameters \(a\) and \(b\) in <a href="./ems_ca_fe_L.html#Lap_1d_sol">Lap_1d_sol</a>), we need to appeal to boundary
|
|
conditions. Concretely, for a finite segment,
|
|
a solution exists and is unique if one is
|
|
provided with any of these possibilities:
|
|
</p>
|
|
|
|
<ul class="org-ul">
|
|
<li>\(\phi\) at both boundaries</li>
|
|
<li>\(\phi\) and \(\frac{d\phi}{dx}\) at one boundary</li>
|
|
<li>\(\phi\) at one boundary, \(\frac{d\phi}{dx}\) at the other.</li>
|
|
</ul>
|
|
|
|
<p>
|
|
Specifying \(\frac{d\phi}{dx}\) at both boundaries
|
|
provides insufficient information, since you get
|
|
an inconsistency if the derivatives don't match.
|
|
</p>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-ems_ca_fe_L_2d" class="outline-6">
|
|
<h6 id="ems_ca_fe_L_2d"><a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a></h6>
|
|
<div class="outline-text-6" id="text-ems_ca_fe_L_2d">
|
|
<p>
|
|
In two dimensions, the potential becomes a function
|
|
of two variables (here: \(x\) and \(y\)), so Laplace's
|
|
equation now reads
|
|
</p>
|
|
<div class="eqlabel" id="orgdc4453f">
|
|
<p>
|
|
<a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><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="org259268b">
|
|
|
|
</div>
|
|
|
|
</div>
|
|
|
|
\begin{equation*}
|
|
\frac{\partial^2 \phi (x,y)}{\partial x^2}
|
|
+ \frac{\partial^2 \phi (x,y)}{\partial y^2} = 0.
|
|
\tag{Lap_2d}\label{Lap_2d}
|
|
\end{equation*}
|
|
|
|
<p>
|
|
Properties:
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>
|
|
<b>Balance</b>: \(\phi(x,y)\) equals the average value around the point:</li>
|
|
</ul>
|
|
<p>
|
|
\[
|
|
\phi(x,y) = \frac{1}{2\pi R} \oint dl ~\phi
|
|
\]
|
|
</p>
|
|
<ul class="org-ul">
|
|
<li>
|
|
<b>No extrema</b>: \(\phi\) has no local maxima or minima. All extrema occur at the boundaries.</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
<div id="outline-container-ems_ca_fe_L_3d" class="outline-6">
|
|
<h6 id="ems_ca_fe_L_3d"><a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a></h6>
|
|
<div class="outline-text-6" id="text-ems_ca_fe_L_3d">
|
|
<p>
|
|
In three dimensions, we will write the potential
|
|
as a function of a 3-dimensional vector, \(\phi({\bf r})\).
|
|
The Laplace equation is (we repeat)
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
{\boldsymbol \nabla}^2 \phi ({\bf r}) = 0
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
<b>Theorem</b>: if \(\phi\) satisfies Laplace, then its value at
|
|
a point equals its value averaged over a sphere
|
|
\(S_R({\bf r})\) of any radius \(R\) centered on this point
|
|
(and of course not containing any charges),
|
|
</p>
|
|
<div class="eqlabel" id="org822a974">
|
|
<p>
|
|
<a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><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="org57d10c8">
|
|
|
|
</div>
|
|
|
|
</div>
|
|
<p>
|
|
\[
|
|
\phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_R({\bf r})} da' ~\phi ({\bf r}')
|
|
\tag{p_ball_avg}\label{p_ball_avg}
|
|
\]
|
|
</p>
|
|
|
|
<details id="org0d796b5">
|
|
<summary id="org6d53cda">
|
|
<strong>Physicist's proof</strong>
|
|
</summary>
|
|
<p>
|
|
Consider a sphere of radius \(R\) centered at the origin
|
|
carrying charge \(q\) spread with a uniform surface charge density over its surface. Bring in a point charge \(q'\) from
|
|
infinity up to a distance \(R'\) (with \(R' > R\)) from the center
|
|
of the sphere.
|
|
</p>
|
|
|
|
<p>
|
|
We know that the field created by the sphere coincides
|
|
with that of a point charge \(q\) at the origin.
|
|
The work required to bring the \(q'\) charge into position is thus
|
|
simply \(W = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.
|
|
</p>
|
|
|
|
<p>
|
|
We can however proceed the other way: fixing \(q'\) in place,
|
|
and then bringing the charged sphere into position;
|
|
the work (energy) has to coincide with our previous result.
|
|
But this energy is now given by the integral of the
|
|
potential \(\phi_{q', {\bf r'}}\)
|
|
created by \(q'\) (sitting at \({\bf r'}\)) over the sphere
|
|
times the surface charge density on the sphere,
|
|
namely
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
W = \oint_{S_R} da ~\sigma ~\phi_{q', {\bf r}'} ({\bf r})
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
But \(\sigma = q/4\pi R^2\) and is a constant over the
|
|
sphere, so \(W = q \times \frac{1}{4\pi R^2} \oint da ~\phi_{q', {\bf r}'} ({\bf r})\).
|
|
</p>
|
|
|
|
<p>
|
|
Equating this with the previous results shows that
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{q'}{4\pi \varepsilon_0 R'} = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})
|
|
\]
|
|
but this also equals the potential at \({\bf r} = 0\) created by the charge
|
|
\(q'\) at \({\bf r'}\), <i>i.e.</i> \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\).
|
|
In other words, we have thus shown that for the potential created by a single point
|
|
charge \(q'\) at \(R'\), the value at a point (here the origin)
|
|
coincides with the value averaged over a sphere
|
|
or an arbitrary radius \(R\) centered on the same point.
|
|
</p>
|
|
|
|
<p>
|
|
By the principle of superposition, this works for an
|
|
arbitrary distribution of charges outside the sphere,
|
|
proving the theorem.
|
|
</p>
|
|
</details>
|
|
|
|
<details id="org68c57fd">
|
|
<summary id="org5db3c01">
|
|
<strong>Formal proof</strong>
|
|
</summary>
|
|
|
|
<p>
|
|
Consider a function \(f({\bf r})\) and its average over
|
|
a ball of radius \(R\) centered on \({\bf r}\):
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R ({\bf r})} da' ~ f ({\bf r}')
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
In spherical coordinates defined around the point \({\bf r}\),
|
|
we have \(da' = R^2 sin \theta d\theta d\varphi \equiv R^2 d\Omega\).
|
|
Differentiating with respect to \(R\),
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi} \oint_{S_R} d\Omega ~\left.\frac{\partial f}{\partial r}\right|_{r=R}
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
with \(f\) differentiated with respect to the radial coordiate.
|
|
We can rewrite this by noting that \(R^2 d\Omega ~\hat{\bf r}\)
|
|
is the normal differential surface area \(d{\bf a}\), while
|
|
\(\left.\frac{\partial f}{\partial r}\right|_{r=R}\) is the radial component of the gradient
|
|
of \(f\) in spherical coordinates. Thus,
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \oint_{S_R} d{\bf a} \cdot ~\nabla f
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
Invoking the divergence theorem and using the definition
|
|
of the Laplacian operator \(\nabla^2 = \nabla \cdot \nabla\),
|
|
we get the following general
|
|
</p>
|
|
|
|
<p>
|
|
<b>Theorem</b>:
|
|
</p>
|
|
<div class="eqlabel" id="org59c453b">
|
|
<p>
|
|
<a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><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="org9481971">
|
|
|
|
</div>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
\[
|
|
\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \int_{V_R} d\tau ~\nabla^2 f
|
|
\tag{dfdR_intLap}\label{dfdR_intLap}
|
|
\]
|
|
</p>
|
|
|
|
|
|
<p>
|
|
For the electrostatic potential away from charges, we have
|
|
\[
|
|
\nabla^2 \phi = 0 ~\rightarrow \frac{d}{dR} \phi_{S_R} = 0
|
|
\]
|
|
namely the ball average is independent of the ball size.
|
|
Since the value at the center is simply the average for
|
|
an infinitesimally small ball, we get the result announced above.
|
|
</p>
|
|
</details>
|
|
|
|
<p>
|
|
<b>Theorem (Earnshaw, mathematical version)</b>: \(\phi\) has no local extrema except at the boundaries.
|
|
</p>
|
|
|
|
<p>
|
|
<b>Proof</b>: write the second derivatives as
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
\frac{\partial^2 \phi({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
|
|
\frac{\partial^2 \phi({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
|
|
\frac{\partial^2 \phi({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
|
|
f_x + f_y + f_z = 0.
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
The \(f_a ({\bf r})\) represent the three components of the curvature of \(\phi({\bf r})\).
|
|
An extremum of \(\phi\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} \phi |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
|
|
for any infinitesimal displacement \(\delta{\bf r}\) around the extremum point. For a local
|
|
minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 \phi}{\partial r_i \partial r_j} \delta r_i \delta r_j > 0\)
|
|
for any displacement vector. Choosing alternately displacements along the three axes,
|
|
the form becomes \(f_x (\delta x)^2\), \(f_y (\delta y)^2\) or \(f_z (\delta z)^2\). Since the squared displacements
|
|
are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f_z > 0\). This is impossible in view
|
|
of the \(f_x + f_y + f_z = 0\) condition above.
|
|
</p>
|
|
|
|
<div class="eqlabel" id="org81bf520">
|
|
<p>
|
|
<a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><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="orgf2a161c">
|
|
|
|
</div>
|
|
|
|
</div>
|
|
<div class="info div" id="orgd9c5641">
|
|
<p>
|
|
<b>Earnshaw's theorem (physical version)</b> <br>
|
|
</p>
|
|
|
|
<p>
|
|
It is impossible to find a static distribution of charges which generates an electrostatic field
|
|
displaying a stable equilibrium position in empty space.
|
|
</p>
|
|
|
|
</div>
|
|
|
|
<p>
|
|
Going back to Poisson's equation, we can make a few comments:
|
|
</p>
|
|
|
|
<ul class="org-ul">
|
|
<li>representation <a href="./ems_es_ep_PL.html#Poi">🐟</a> highlights the 'local' nature of the coupling between electrostatic fields and charges: fields are 'created' where the charges 'sit'. This is also seen by looking at the integrand of <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>. If electrostatics was nonlocal, a modified representation like <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> would still exist, but not a local differential one like Poisson's equation.</li>
|
|
|
|
<li>as written, representations <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a> and <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> require the knowledge of the charge density distribution \(\rho({\bf r})\) throughout space to determine the potential at any given point.</li>
|
|
|
|
<li>Poisson's equation <a href="./ems_es_ep_PL.html#Poi">🐟</a>, being purely local, might allow to determine the potential at a specified point, provided we know the charge density distribution around this specified point, and at some set of other reference points (to make the solution unique).</li>
|
|
</ul>
|
|
|
|
|
|
<p>
|
|
We therefore want to ask the question: <i>under what conditions can an electrostatic problem be fully
|
|
defined by solving Poisson's equation ?</i>
|
|
</p>
|
|
|
|
<p>
|
|
We start by mentioning some cases, and interpreting them thereafter.
|
|
</p>
|
|
|
|
<p>
|
|
<b>Charge density is known throughout space</b>: in this case,
|
|
the electrostatic potential is uniquely determined
|
|
by Poisson's equation, which
|
|
is explicitly solved by <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>.
|
|
One can explicitly verify this:
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
{\boldsymbol \nabla}^2 \phi ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
|
|
= \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}.
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
where we have used <a href="./c_m_dd_3d.html#Lap1or">Lap1or</a>, and the fact that the delta function is always resolved since we
|
|
integrate over all space. Note: it is implicitly assumed that the integral in <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
|
|
converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved.
|
|
</p>
|
|
|
|
<p>
|
|
<b>"Known boundary charge" case: charge density in closed volume and boundary surface charge density are both known</b>: the electrostatic potential is uniquely determined
|
|
in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
|
|
\(\rho ({\bf r})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),
|
|
and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\).
|
|
Of course, \({\cal S}\) need not be a connected surface.
|
|
</p>
|
|
|
|
|
|
<p>
|
|
<b>"Known boundary potential" case: charge density in closed volume and potential at boundary are both known</b>: the electrostatic potential is uniquely determined
|
|
in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
|
|
\(\rho ({\bf r})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the
|
|
boundary \({\cal S}\). Of course, \({\cal S}\) need not be a connected surface.
|
|
</p>
|
|
|
|
<p>
|
|
Here, the logic is quite simple: since the electrostatic potential is known on all the surface enclosing
|
|
the space \({\cal V}\), and since Poisson's equation is local, we need not consider anything outside of \({\cal V}\)
|
|
to obtain \(V\) within \({\cal V}\).
|
|
</p>
|
|
|
|
<p>
|
|
Given a solution \(V_1 ({\bf r})\), we can easily show that it is unique. Suppose there was another solution
|
|
\(V_2 ({\bf r})\). Look at the difference, \(U \equiv V_1 - V_2\). In the bulk, \(U\) obeys the Laplace
|
|
equation
|
|
</p>
|
|
|
|
<p>
|
|
\[
|
|
{\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0.
|
|
\]
|
|
</p>
|
|
|
|
<p>
|
|
Moreover, \(U ({\bf r}) = 0\) for \({\bf r} \in {\cal S}\). Since solutions to the Laplace equation take
|
|
their maximal and minimal value on the boundary, we must have \(U = 0\) \(\forall {\bf r} \in {\cal V}\)
|
|
</p>
|
|
|
|
|
|
<p>
|
|
This all feels a bit amateurish and not very systematic. Can we be more precise and general? What kinds of boundary information do we really need to specify the solution uniquely ?
|
|
</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
|
|
|
|
<br><ul class="navigation-links"><li>Prev: <a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential <small>[ems.ca.fe]</small></a></li><li>Next: <a href="ems_ca_fe_g.html">Green's Identities <small>[ems.ca.fe.g]</small></a></li><li>Up: <a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential <small>[ems.ca.fe]</small></a></li></ul>
|
|
<br>
|
|
<hr>
|
|
<div class="license">
|
|
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
|
|
target="_blank" class="m-2">
|
|
<img alt="Creative Commons License" style="border-width:0"
|
|
src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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</a>
|
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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: 2022-03-01 Tue 08:14</p>
|
|
<p class="validation"></p>
|
|
</div>
|
|
|
|
|
|
</div>
|
|
</html> |