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  239. 

  240. <div id="content">

  241. <header>

  242. <h1 class="title">

  243. <a href="./index.html" class="homepage-link">Pre-Quantum Electrodynamics</a>

  244. </h1>

  245. </header>

  246. <nav id="collapsed-table-of-contents">

  247. <details>

  248. <summary>

  249. Table of contents

  250. </summary>

  251. <ul>

  252. <li>

  253. 

  254. <details>

  255. <summary>

  256. <a href="./in.html#in">Introduction</a><span class="headline-id">in</span>

  257. 

  258. 

  259. </summary>

  260. <ul>

  261. <li>

  262. <a href="./in_p.html#in_p">Preface</a><span class="headline-id">in.p</span>

  263. 

  264. </li>

  265. <li>

  266. 

  267. <details>

  268. <summary>

  269. <a href="./in_t.html#in_t">Tips for the reader</a><span class="headline-id">in.t</span>

  270. 

  271. 

  272. </summary>

  273. <ul>

  274. <li>

  275. <a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>

  276. 

  277. </li>

  278. <li>

  279. <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>

  280. 

  281. </li>

  282. 

  283. </ul>

  284. </details>

  285. </li>

  286. 

  287. </ul>

  288. </details>

  289. </li>

  290. <li>

  291. 

  292. <details open="">

  293. <summary class="toc-open">

  294. <a href="./ems.html#ems">Electromagnetostatics</a><span class="headline-id">ems</span>

  295. 

  296. 

  297. </summary>

  298. <ul>

  299. <li>

  300. 

  301. <details>

  302. <summary>

  303. <a href="./ems_es.html#ems_es">Electrostatics</a><span class="headline-id">ems.es</span>

  304. 

  305. 

  306. </summary>

  307. <ul>

  308. <li>

  309. 

  310. <details>

  311. <summary>

  312. <a href="./ems_es_ec.html#ems_es_ec">Electric Charge</a><span class="headline-id">ems.es.ec</span>

  313. 

  314. 

  315. </summary>

  316. <ul>

  317. <li>

  318. <a href="./ems_es_ec_b.html#ems_es_ec_b">Basics</a><span class="headline-id">ems.es.ec.b</span>

  319. 

  320. </li>

  321. <li>

  322. <a href="./ems_es_ec_c.html#ems_es_ec_c">Conservation</a><span class="headline-id">ems.es.ec.c</span>

  323. 

  324. </li>

  325. <li>

  326. <a href="./ems_es_ec_q.html#ems_es_ec_q">Quantization</a><span class="headline-id">ems.es.ec.q</span>

  327. 

  328. </li>

  329. <li>

  330. <a href="./ems_es_ec_s.html#ems_es_ec_s">Structure</a><span class="headline-id">ems.es.ec.s</span>

  331. 

  332. </li>

  333. 

  334. </ul>

  335. </details>

  336. </li>

  337. <li>

  338. 

  339. <details>

  340. <summary>

  341. <a href="./ems_es_efo.html#ems_es_efo">Electric Force and Energy</a><span class="headline-id">ems.es.efo</span>

  342. 

  343. 

  344. </summary>

  345. <ul>

  346. <li>

  347. <a href="./ems_es_efo_cl.html#ems_es_efo_cl">Coulomb's Law</a><span class="headline-id">ems.es.efo.cl</span>

  348. 

  349. </li>

  350. <li>

  351. <a href="./ems_es_efo_ps.html#ems_es_efo_ps">Principle of Superposition</a><span class="headline-id">ems.es.efo.ps</span>

  352. 

  353. </li>

  354. <li>

  355. <a href="./ems_es_efo_exp.html#ems_es_efo_exp">Experimental Investigations</a><span class="headline-id">ems.es.efo.exp</span>

  356. 

  357. </li>

  358. <li>

  359. <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>

  360. 

  361. </li>

  362. 

  363. </ul>

  364. </details>

  365. </li>

  366. <li>

  367. 

  368. <details>

  369. <summary>

  370. <a href="./ems_es_ef.html#ems_es_ef">Electrostatic Fields</a><span class="headline-id">ems.es.ef</span>

  371. 

  372. 

  373. </summary>

  374. <ul>

  375. <li>

  376. <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>

  377. 

  378. </li>

  379. <li>

  380. <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>

  381. 

  382. </li>

  383. <li>

  384. <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>

  385. 

  386. </li>

  387. <li>

  388. <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>

  389. 

  390. </li>

  391. 

  392. </ul>

  393. </details>

  394. </li>

  395. <li>

  396. 

  397. <details>

  398. <summary>

  399. <a href="./ems_es_ep.html#ems_es_ep">The Electrostatic Potential</a><span class="headline-id">ems.es.ep</span>

  400. 

  401. 

  402. </summary>

  403. <ul>

  404. <li>

  405. <a href="./ems_es_ep_d.html#ems_es_ep_d">Definition</a><span class="headline-id">ems.es.ep.d</span>

  406. 

  407. </li>

  408. <li>

  409. <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>

  410. 

  411. </li>

  412. <li>

  413. <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>

  414. 

  415. </li>

  416. <li>

  417. <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>

  418. 

  419. </li>

  420. <li>

  421. <a href="./ems_es_ep_bc.html#ems_es_ep_bc">Electrostatic Boundary Conditions</a><span class="headline-id">ems.es.ep.bc</span>

  422. 

  423. </li>

  424. 

  425. </ul>

  426. </details>

  427. </li>

  428. <li>

  429. <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>

  430. 

  431. </li>

  432. <li>

  433. 

  434. <details>

  435. <summary>

  436. <a href="./ems_es_c.html#ems_es_c">Conductors</a><span class="headline-id">ems.es.c</span>

  437. 

  438. 

  439. </summary>

  440. <ul>

  441. <li>

  442. <a href="./ems_es_c_p.html#ems_es_c_p">Properties</a><span class="headline-id">ems.es.c.p</span>

  443. 

  444. </li>

  445. <li>

  446. <a href="./ems_es_c_ic.html#ems_es_c_ic">Induced Charges</a><span class="headline-id">ems.es.c.ic</span>

  447. 

  448. </li>

  449. <li>

  450. <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>

  451. 

  452. </li>

  453. <li>

  454. <a href="./ems_es_c_cap.html#ems_es_c_cap">Capacitors</a><span class="headline-id">ems.es.c.cap</span>

  455. 

  456. </li>

  457. 

  458. </ul>

  459. </details>

  460. </li>

  461. 

  462. </ul>

  463. </details>

  464. </li>

  465. <li>

  466. 

  467. <details open="">

  468. <summary class="toc-open">

  469. <a href="./ems_ca.html#ems_ca">Calculating or Approximating the Electrostatic Potential</a><span class="headline-id">ems.ca</span>

  470. 

  471. 

  472. </summary>

  473. <ul>

  474. <li>

  475. 

  476. <details open="">

  477. <summary class="toc-open">

  478. <a href="./ems_ca_fe.html#ems_ca_fe">Fundamental Equations for the Electrostatic Potential</a><span class="headline-id">ems.ca.fe</span>

  479. 

  480. 

  481. </summary>

  482. <ul>

  483. <li class="toc-currentpage">

  484. <a href="./ems_ca_fe_L.html#ems_ca_fe_L">The Laplace Equation</a><span class="headline-id">ems.ca.fe.L</span>

  485. 

  486. </li>

  487. <li>

  488. <a href="./ems_ca_fe_g.html#ems_ca_fe_g">Green's Identities</a><span class="headline-id">ems.ca.fe.g</span>

  489. 

  490. </li>

  491. <li>

  492. <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>

  493. 

  494. </li>

  495. 

  496. </ul>

  497. </details>

  498. </li>

  499. <li>

  500. 

  501. <details>

  502. <summary>

  503. <a href="./ems_ca_mi.html#ems_ca_mi">The Method of Images</a><span class="headline-id">ems.ca.mi</span>

  504. 

  505. 

  506. </summary>

  507. <ul>

  508. <li>

  509. <a href="./ems_ca_mi_isc.html#ems_ca_mi_isc">Induced Surface Charges</a><span class="headline-id">ems.ca.mi.isc</span>

  510. 

  511. </li>

  512. <li>

  513. <a href="./ems_ca_mi_fe.html#ems_ca_mi_fe">Force and Energy</a><span class="headline-id">ems.ca.mi.fe</span>

  514. 

  515. </li>

  516. <li>

  517. <a href="./ems_ca_mi_o.html#ems_ca_mi_o">Other Image Problems</a><span class="headline-id">ems.ca.mi.o</span>

  518. 

  519. </li>

  520. 

  521. </ul>

  522. </details>

  523. </li>

  524. <li>

  525. 

  526. <details>

  527. <summary>

  528. <a href="./ems_ca_sv.html#ems_ca_sv">Separation of Variables</a><span class="headline-id">ems.ca.sv</span>

  529. 

  530. 

  531. </summary>

  532. <ul>

  533. <li>

  534. <a href="./ems_ca_sv_car.html#ems_ca_sv_car">Cartesian Coordinates</a><span class="headline-id">ems.ca.sv.car</span>

  535. 

  536. </li>

  537. <li>

  538. <a href="./ems_ca_sv_cyl.html#ems_ca_sv_cyl">Cylindrical Coordinates</a><span class="headline-id">ems.ca.sv.cyl</span>

  539. 

  540. </li>

  541. <li>

  542. <a href="./ems_ca_sv_sph.html#ems_ca_sv_sph">Spherical Coordinates</a><span class="headline-id">ems.ca.sv.sph</span>

  543. 

  544. </li>

  545. 

  546. </ul>

  547. </details>

  548. </li>

  549. <li>

  550. 

  551. <details>

  552. <summary>

  553. <a href="./ems_ca_me.html#ems_ca_me">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span>

  554. 

  555. 

  556. </summary>

  557. <ul>

  558. <li>

  559. <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>

  560. 

  561. </li>

  562. <li>

  563. <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>

  564. 

  565. </li>

  566. <li>

  567. <a href="./ems_ca_me_h.html#ems_ca_me_h">Higher Moments</a><span class="headline-id">ems.ca.me.h</span>

  568. 

  569. </li>

  570. <li>

  571. <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>

  572. 

  573. </li>

  574. <li>

  575. <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>

  576. 

  577. </li>

  578. 

  579. </ul>

  580. </details>

  581. </li>

  582. 

  583. </ul>

  584. </details>

  585. </li>

  586. <li>

  587. 

  588. <details>

  589. <summary>

  590. <a href="./ems_ms.html#ems_ms">Magnetostatics</a><span class="headline-id">ems.ms</span>

  591. 

  592. 

  593. </summary>

  594. <ul>

  595. <li>

  596. 

  597. <details>

  598. <summary>

  599. <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>

  600. 

  601. 

  602. </summary>

  603. <ul>

  604. <li>

  605. <a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>

  606. 

  607. </li>

  608. <li>

  609. <a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>

  610. 

  611. </li>

  612. 

  613. </ul>

  614. </details>

  615. </li>

  616. <li>

  617. <a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>

  618. 

  619. </li>

  620. <li>

  621. <a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>

  622. 

  623. </li>

  624. <li>

  625. 

  626. <details>

  627. <summary>

  628. <a href="./ems_ms_dcB.html#ems_ms_dcB">Divergence and Curl of \({\bf B}\)</a><span class="headline-id">ems.ms.dcB</span>

  629. 

  630. 

  631. </summary>

  632. <ul>

  633. <li>

  634. <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>

  635. 

  636. </li>

  637. <li>

  638. <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>

  639. 

  640. </li>

  641. <li>

  642. <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>

  643. 

  644. </li>

  645. 

  646. </ul>

  647. </details>

  648. </li>

  649. <li>

  650. 

  651. <details>

  652. <summary>

  653. <a href="./ems_ms_vp.html#ems_ms_vp">The Vector Potential</a><span class="headline-id">ems.ms.vp</span>

  654. 

  655. 

  656. </summary>

  657. <ul>

  658. <li>

  659. <a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>

  660. 

  661. </li>

  662. <li>

  663. <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>

  664. 

  665. </li>

  666. <li>

  667. <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>

  668. 

  669. </li>

  670. <li>

  671. <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>

  672. 

  673. </li>

  674. <li>

  675. <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>

  676. 

  677. </li>

  678. 

  679. </ul>

  680. </details>

  681. </li>

  682. 

  683. </ul>

  684. </details>

  685. </li>

  686. 

  687. </ul>

  688. </details>

  689. </li>

  690. <li>

  691. 

  692. <details>

  693. <summary>

  694. <a href="./emsm.html#emsm">Electromagnetostatics in matter</a><span class="headline-id">emsm</span>

  695. 

  696. 

  697. </summary>

  698. <ul>

  699. <li>

  700. 

  701. <details>

  702. <summary>

  703. <a href="./emsm_esm.html#emsm_esm">Electrostatics in matter</a><span class="headline-id">emsm.esm</span>

  704. 

  705. 

  706. </summary>

  707. <ul>

  708. <li>

  709. 

  710. <details>

  711. <summary>

  712. <a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>

  713. 

  714. 

  715. </summary>

  716. <ul>

  717. <li>

  718. <a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>

  719. 

  720. </li>

  721. <li>

  722. <a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</span>

  723. 

  724. </li>

  725. 

  726. </ul>

  727. </details>

  728. </li>

  729. <li>

  730. 

  731. <details>

  732. <summary>

  733. <a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>

  734. 

  735. 

  736. </summary>

  737. <ul>

  738. <li>

  739. <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>

  740. 

  741. </li>

  742. <li>

  743. <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>

  744. 

  745. </li>

  746. 

  747. </ul>

  748. </details>

  749. </li>

  750. <li>

  751. 

  752. <details>

  753. <summary>

  754. <a href="./emsm_esm_D.html#emsm_esm_D">The Electric Displacement</a><span class="headline-id">emsm.esm.D</span>

  755. 

  756. 

  757. </summary>

  758. <ul>

  759. <li>

  760. <a href="./emsm_esm_D_bc.html#emsm_esm_D_bc">Boundary Conditions</a><span class="headline-id">emsm.esm.D.bc</span>

  761. 

  762. </li>

  763. 

  764. </ul>

  765. </details>

  766. </li>

  767. <li>

  768. <a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>

  769. 

  770. </li>

  771. <li>

  772. 

  773. <details>

  774. <summary>

  775. <a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>

  776. 

  777. 

  778. </summary>

  779. <ul>

  780. <li>

  781. <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>

  782. 

  783. </li>

  784. <li>

  785. <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>

  786. 

  787. </li>

  788. <li>

  789. <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>

  790. 

  791. </li>

  792. <li>

  793. <a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>

  794. 

  795. </li>

  796. 

  797. </ul>

  798. </details>

  799. </li>

  800. 

  801. </ul>

  802. </details>

  803. </li>

  804. <li>

  805. 

  806. <details>

  807. <summary>

  808. <a href="./emsm_msm.html#emsm_msm">Magnetostatics in matter</a><span class="headline-id">emsm.msm</span>

  809. 

  810. 

  811. </summary>

  812. <ul>

  813. <li>

  814. 

  815. <details>

  816. <summary>

  817. <a href="./emsm_msm_m.html#emsm_msm_m">Magnetization</a><span class="headline-id">emsm.msm.m</span>

  818. 

  819. 

  820. </summary>

  821. <ul>

  822. <li>

  823. <a href="./emsm_msm_m_dpf.html#emsm_msm_m_dpf">Diamagnetism, Paramagnetism, Ferromagnetism</a><span class="headline-id">emsm.msm.m.dpf</span>

  824. 

  825. </li>

  826. <li>

  827. <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>

  828. 

  829. </li>

  830. <li>

  831. <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>

  832. 

  833. </li>

  834. 

  835. </ul>

  836. </details>

  837. </li>

  838. <li>

  839. 

  840. <details>

  841. <summary>

  842. <a href="./emsm_msm_fmo.html#emsm_msm_fmo">The Field of a Magnetized Object</a><span class="headline-id">emsm.msm.fmo</span>

  843. 

  844. 

  845. </summary>

  846. <ul>

  847. <li>

  848. <a href="./emsm_msm_fmo_bc.html#emsm_msm_fmo_bc">Bound Currents</a><span class="headline-id">emsm.msm.fmo.bc</span>

  849. 

  850. </li>

  851. <li>

  852. <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>

  853. 

  854. </li>

  855. <li>

  856. <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>

  857. 

  858. </li>

  859. 

  860. </ul>

  861. </details>

  862. </li>

  863. <li>

  864. 

  865. <details>

  866. <summary>

  867. <a href="./emsm_msm_H.html#emsm_msm_H">The H Field</a><span class="headline-id">emsm.msm.H</span>

  868. 

  869. 

  870. </summary>

  871. <ul>

  872. <li>

  873. <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>

  874. 

  875. </li>

  876. 

  877. </ul>

  878. </details>

  879. </li>

  880. <li>

  881. 

  882. <details>

  883. <summary>

  884. <a href="./emsm_msm_lnlm.html#emsm_msm_lnlm">Linear and Nonlinear Media</a><span class="headline-id">emsm.msm.lnlm</span>

  885. 

  886. 

  887. </summary>

  888. <ul>

  889. <li>

  890. <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>

  891. 

  892. </li>

  893. <li>

  894. <a href="./emsm_msm_lnlm_fm.html#emsm_msm_lnlm_fm">Ferromagnetism</a><span class="headline-id">emsm.msm.lnlm.fm</span>

  895. 

  896. </li>

  897. 

  898. </ul>

  899. </details>

  900. </li>

  901. 

  902. </ul>

  903. </details>

  904. </li>

  905. 

  906. </ul>

  907. </details>

  908. </li>

  909. <li>

  910. 

  911. <details>

  912. <summary>

  913. <a href="./emd.html#emd">Electromagnetodynamics</a><span class="headline-id">emd</span>

  914. 

  915. 

  916. </summary>

  917. <ul>

  918. <li>

  919. 

  920. <details>

  921. <summary>

  922. <a href="./emd_Fl.html#emd_Fl">Induction: Faraday's Law</a><span class="headline-id">emd.Fl</span>

  923. 

  924. 

  925. </summary>

  926. <ul>

  927. <li>

  928. <a href="./emd_Fl_Fl.html#emd_Fl_Fl">Faraday's Law</a><span class="headline-id">emd.Fl.Fl</span>

  929. 

  930. </li>

  931. <li>

  932. <a href="./emd_Fl_ief.html#emd_Fl_ief">The Induced Electric Field</a><span class="headline-id">emd.Fl.ief</span>

  933. 

  934. </li>

  935. <li>

  936. <a href="./emd_Fl_i.html#emd_Fl_i">Inductance</a><span class="headline-id">emd.Fl.i</span>

  937. 

  938. </li>

  939. <li>

  940. <a href="./emd_Fl_e.html#emd_Fl_e">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span>

  941. 

  942. </li>

  943. 

  944. </ul>

  945. </details>

  946. </li>

  947. <li>

  948. 

  949. <details>

  950. <summary>

  951. <a href="./emd_Me.html#emd_Me">Maxwell's Equations</a><span class="headline-id">emd.Me</span>

  952. 

  953. 

  954. </summary>

  955. <ul>

  956. <li>

  957. <a href="./emd_Me_ebM.html#emd_Me_ebM">Electrodynamics Before Maxwell</a><span class="headline-id">emd.Me.ebM</span>

  958. 

  959. </li>

  960. <li>

  961. <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>

  962. 

  963. </li>

  964. <li>

  965. <a href="./emd_Me_Me.html#emd_Me_Me">Maxwell's Equations</a><span class="headline-id">emd.Me.Me</span>

  966. 

  967. </li>

  968. <li>

  969. <a href="./emd_Me_mc.html#emd_Me_mc">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span>

  970. 

  971. </li>

  972. 

  973. </ul>

  974. </details>

  975. </li>

  976. <li>

  977. 

  978. <details>

  979. <summary>

  980. <a href="./emd_ce.html#emd_ce">Charge and Energy Flows</a><span class="headline-id">emd.ce</span>

  981. 

  982. 

  983. </summary>

  984. <ul>

  985. <li>

  986. <a href="./emd_ce_ce.html#emd_ce_ce">The Continuity Equation</a><span class="headline-id">emd.ce.ce</span>

  987. 

  988. </li>

  989. <li>

  990. <a href="./emd_ce_poy.html#emd_ce_poy">Poynting's Theorem; the Poynting Vector</a><span class="headline-id">emd.ce.poy</span>

  991. 

  992. </li>

  993. <li>

  994. <a href="./emd_ce_mst.html#emd_ce_mst">Maxwell's Stress Tensor</a><span class="headline-id">emd.ce.mst</span>

  995. 

  996. </li>

  997. <li>

  998. <a href="./emd_ce_mom.html#emd_ce_mom">Momentum</a><span class="headline-id">emd.ce.mom</span>

  999. 

  1000. </li>

  1001. <li>

  1002. <a href="./emd_ce_amom.html#emd_ce_amom">Angular Momentum</a><span class="headline-id">emd.ce.amom</span>

  1003. 

  1004. </li>

  1005. 

  1006. </ul>

  1007. </details>

  1008. </li>

  1009. <li>

  1010. 

  1011. <details>

  1012. <summary>

  1013. <a href="./emd_emw.html#emd_emw">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span>

  1014. 

  1015. 

  1016. </summary>

  1017. <ul>

  1018. <li>

  1019. <a href="./emd_emw_we.html#emd_emw_we">The Wave Equation</a><span class="headline-id">emd.emw.we</span>

  1020. 

  1021. </li>

  1022. <li>

  1023. <a href="./emd_emw_mpw.html#emd_emw_mpw">Monochromatic Plane Waves</a><span class="headline-id">emd.emw.mpw</span>

  1024. 

  1025. </li>

  1026. <li>

  1027. <a href="./emd_emw_ep.html#emd_emw_ep">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span>

  1028. 

  1029. </li>

  1030. 

  1031. </ul>

  1032. </details>

  1033. </li>

  1034. 

  1035. </ul>

  1036. </details>

  1037. </li>

  1038. <li>

  1039. 

  1040. <details>

  1041. <summary>

  1042. <a href="./emdm.html#emdm">Electromagnetodynamics in Matter</a><span class="headline-id">emdm</span>

  1043. 

  1044. 

  1045. </summary>

  1046. <ul>

  1047. <li>

  1048. 

  1049. <details>

  1050. <summary>

  1051. <a href="./emdm_Me.html#emdm_Me">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me</span>

  1052. 

  1053. 

  1054. </summary>

  1055. <ul>

  1056. <li>

  1057. <a href="./emdm_Me_Mem.html#emdm_Me_Mem">Maxwell's Equations in Matter</a><span class="headline-id">emdm.Me.Mem</span>

  1058. 

  1059. </li>

  1060. <li>

  1061. <a href="./emdm_Me_bc.html#emdm_Me_bc">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span>

  1062. 

  1063. </li>

  1064. 

  1065. </ul>

  1066. </details>

  1067. </li>

  1068. <li>

  1069. 

  1070. <details>

  1071. <summary>

  1072. <a href="./emdm_emwm.html#emdm_emwm">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span>

  1073. 

  1074. 

  1075. </summary>

  1076. <ul>

  1077. <li>

  1078. <a href="./emdm_emwm_plm.html#emdm_emwm_plm">Propagation in Linear Media</a><span class="headline-id">emdm.emwm.plm</span>

  1079. 

  1080. </li>

  1081. <li>

  1082. <a href="./emdm_emwm_refr.html#emdm_emwm_refr">Refraction</a><span class="headline-id">emdm.emwm.refr</span>

  1083. 

  1084. </li>

  1085. <li>

  1086. 

  1087. <details>

  1088. <summary>

  1089. <a href="./emdm_emwm_refl.html#emdm_emwm_refl">Reflection and Transmission</a><span class="headline-id">emdm.emwm.refl</span>

  1090. 

  1091. 

  1092. </summary>

  1093. <ul>

  1094. <li>

  1095. <a href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span>

  1096. 

  1097. </li>

  1098. <li>

  1099. <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>

  1100. 

  1101. </li>

  1102. 

  1103. </ul>

  1104. </details>

  1105. </li>

  1106. <li>

  1107. 

  1108. <details>

  1109. <summary>

  1110. <a href="./emdm_emwm_ad.html#emdm_emwm_ad">Absorption and Dispersion</a><span class="headline-id">emdm.emwm.ad</span>

  1111. 

  1112. 

  1113. </summary>

  1114. <ul>

  1115. <li>

  1116. <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>

  1117. 

  1118. </li>

  1119. 

  1120. </ul>

  1121. </details>

  1122. </li>

  1123. <li>

  1124. 

  1125. <details>

  1126. <summary>

  1127. <a href="./emdm_emwm_wg.html#emdm_emwm_wg">Waveguides</a><span class="headline-id">emdm.emwm.wg</span>

  1128. 

  1129. 

  1130. </summary>

  1131. <ul>

  1132. <li>

  1133. <a href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw">Guided waves</a><span class="headline-id">emdm.emwm.wg.gw</span>

  1134. 

  1135. </li>

  1136. <li>

  1137. <a href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r">Rectangular Waveguides</a><span class="headline-id">emdm.emwm.wg.r</span>

  1138. 

  1139. </li>

  1140. <li>

  1141. <a href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span>

  1142. 

  1143. </li>

  1144. 

  1145. </ul>

  1146. </details>

  1147. </li>

  1148. 

  1149. </ul>

  1150. </details>

  1151. </li>

  1152. 

  1153. </ul>

  1154. </details>

  1155. </li>

  1156. <li>

  1157. 

  1158. <details>

  1159. <summary>

  1160. <a href="./emf.html#emf">Electromagnetic Fields</a><span class="headline-id">emf</span>

  1161. 

  1162. 

  1163. </summary>

  1164. <ul>

  1165. <li>

  1166. <a href="./emf_svp.html#emf_svp">Scalar and Vector Potentials</a><span class="headline-id">emf.svp</span>

  1167. 

  1168. </li>

  1169. <li>

  1170. 

  1171. <details>

  1172. <summary>

  1173. <a href="./emf_g.html#emf_g">Gauge Freedom and Choices</a><span class="headline-id">emf.g</span>

  1174. 

  1175. 

  1176. </summary>

  1177. <ul>

  1178. <li>

  1179. <a href="./emf_g_Cg.html#emf_g_Cg">Coulomb Gauge</a><span class="headline-id">emf.g.Cg</span>

  1180. 

  1181. </li>

  1182. <li>

  1183. <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>

  1184. 

  1185. </li>

  1186. 

  1187. </ul>

  1188. </details>

  1189. </li>

  1190. 

  1191. </ul>

  1192. </details>

  1193. </li>

  1194. <li>

  1195. 

  1196. <details>

  1197. <summary>

  1198. <a href="./red.html#red">Relativistic Electrodynamics</a><span class="headline-id">red</span>

  1199. 

  1200. 

  1201. </summary>

  1202. <ul>

  1203. <li>

  1204. 

  1205. <details>

  1206. <summary>

  1207. <a href="./red_sr.html#red_sr">Special Relativity</a><span class="headline-id">red.sr</span>

  1208. 

  1209. 

  1210. </summary>

  1211. <ul>

  1212. <li>

  1213. <a href="./red_sr_p.html#red_sr_p">Postulates and their consequences</a><span class="headline-id">red.sr.p</span>

  1214. 

  1215. </li>

  1216. <li>

  1217. <a href="./red_sr_Lt.html#red_sr_Lt">Lorentz Transformations</a><span class="headline-id">red.sr.Lt</span>

  1218. 

  1219. </li>

  1220. <li>

  1221. <a href="./red_sr_4v.html#red_sr_4v">Covariant and Contravariant Four-Vectors</a><span class="headline-id">red.sr.4v</span>

  1222. 

  1223. </li>

  1224. 

  1225. </ul>

  1226. </details>

  1227. </li>

  1228. <li>

  1229. 

  1230. <details>

  1231. <summary>

  1232. <a href="./red_rm.html#red_rm">Relativistic Mechanics</a><span class="headline-id">red.rm</span>

  1233. 

  1234. 

  1235. </summary>

  1236. <ul>

  1237. <li>

  1238. <a href="./red_rm_pt.html#red_rm_pt">Proper Time and Proper Velocity</a><span class="headline-id">red.rm.pt</span>

  1239. 

  1240. </li>

  1241. <li>

  1242. <a href="./red_rm_rme.html#red_rm_rme">Relativistic Momentum and Energy</a><span class="headline-id">red.rm.rme</span>

  1243. 

  1244. </li>

  1245. <li>

  1246. <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>

  1247. 

  1248. </li>

  1249. 

  1250. </ul>

  1251. </details>

  1252. </li>

  1253. <li>

  1254. 

  1255. <details>

  1256. <summary>

  1257. <a href="./red_rem.html#red_rem">Relativistic Electromagnetism</a><span class="headline-id">red.rem</span>

  1258. 

  1259. 

  1260. </summary>

  1261. <ul>

  1262. <li>

  1263. <a href="./red_rem_mre.html#red_rem_mre">Magnetism as a Relativistic Effect</a><span class="headline-id">red.rem.mre</span>

  1264. 

  1265. </li>

  1266. <li>

  1267. <a href="./red_rem_Ltf.html#red_rem_Ltf">Lorentz Transformation of Electromagnetic Fields</a><span class="headline-id">red.rem.Ltf</span>

  1268. 

  1269. </li>

  1270. <li>

  1271. <a href="./red_rem_Fmunu.html#red_rem_Fmunu">The Field Tensor</a><span class="headline-id">red.rem.Fmunu</span>

  1272. 

  1273. </li>

  1274. <li>

  1275. <a href="./red_rem_Me.html#red_rem_Me">Maxwell's Equations in Relativistic Notation</a><span class="headline-id">red.rem.Me</span>

  1276. 

  1277. </li>

  1278. 

  1279. </ul>

  1280. </details>

  1281. </li>

  1282. 

  1283. </ul>

  1284. </details>

  1285. </li>

  1286. <li>

  1287. 

  1288. <details>

  1289. <summary>

  1290. <a href="./qed.html#qed">Quantum Electrodynamics</a><span class="headline-id">qed</span>

  1291. 

  1292. 

  1293. </summary>

  1294. <ul>

  1295. <li>

  1296. <a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>

  1297. 

  1298. </li>

  1299. 

  1300. </ul>

  1301. </details>

  1302. </li>

  1303. <li>

  1304. 

  1305. <details>

  1306. <summary>

  1307. <a href="./d.html#d">Diagnostics</a><span class="headline-id">d</span>

  1308. 

  1309. 

  1310. </summary>

  1311. <ul>

  1312. <li>

  1313. <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>

  1314. 

  1315. </li>

  1316. <li>

  1317. <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>

  1318. 

  1319. </li>

  1320. <li>

  1321. <a href="./d_emsm.html#d_emsm">Diagnostics: Electromagnetostatics in Matter</a><span class="headline-id">d.emsm</span>

  1322. 

  1323. </li>

  1324. <li>

  1325. <a href="./d_ems_ms.html#d_ems_ms">Diagnostics: Magnetostatics</a><span class="headline-id">d.ems.ms</span>

  1326. 

  1327. </li>

  1328. <li>

  1329. <a href="./d_emsm_msm.html#d_emsm_msm">Diagnostics: Magnetostatics in Matter</a><span class="headline-id">d.emsm.msm</span>

  1330. 

  1331. </li>

  1332. <li>

  1333. <a href="./d_emd.html#d_emd">Diagnostics: Electromagnetodynamics</a><span class="headline-id">d.emd</span>

  1334. 

  1335. </li>

  1336. <li>

  1337. <a href="./d_emd_ce.html#d_emd_ce">Diagnostics: Conservation Laws</a><span class="headline-id">d.emd.ce</span>

  1338. 

  1339. </li>

  1340. <li>

  1341. <a href="./d_emd_emw.html#d_emd_emw">Diagnostics: Electromagnetic Waves</a><span class="headline-id">d.emd.emw</span>

  1342. 

  1343. </li>

  1344. <li>

  1345. <a href="./d_emf.html#d_emf">Diagnostics: Potentials, Gauges and Fields</a><span class="headline-id">d.emf</span>

  1346. 

  1347. </li>

  1348. <li>

  1349. <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>

  1350. 

  1351. </li>

  1352. <li>

  1353. <a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>

  1354. 

  1355. </li>

  1356. 

  1357. </ul>

  1358. </details>

  1359. </li>

  1360. <li>

  1361. 

  1362. <details>

  1363. <summary>

  1364. <a href="./a.html#a">Appendices</a><span class="headline-id">a</span>

  1365. 

  1366. 

  1367. </summary>

  1368. <ul>

  1369. <li>

  1370. <a href="./a_l.html#a_l">Literature</a><span class="headline-id">a.l</span>

  1371. 

  1372. </li>

  1373. 

  1374. </ul>

  1375. </details>

  1376. </li>

  1377. <li>

  1378. 

  1379. <details>

  1380. <summary>

  1381. <a href="./c.html#c">Compendium</a><span class="headline-id">c</span>

  1382. 

  1383. 

  1384. </summary>

  1385. <ul>

  1386. <li>

  1387. 

  1388. <details>

  1389. <summary>

  1390. <a href="./c_m.html#c_m">Mathematics</a><span class="headline-id">c.m</span>

  1391. 

  1392. 

  1393. </summary>

  1394. <ul>

  1395. <li>

  1396. 

  1397. <details>

  1398. <summary>

  1399. <a href="./c_m_va.html#c_m_va">Vector Analysis</a><span class="headline-id">c.m.va</span>

  1400. 

  1401. 

  1402. </summary>

  1403. <ul>

  1404. <li>

  1405. <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>

  1406. 

  1407. </li>

  1408. <li>

  1409. <a href="./c_m_va_sp.html#c_m_va_sp">Scalar product</a><span class="headline-id">c.m.va.sp</span>

  1410. 

  1411. </li>

  1412. <li>

  1413. <a href="./c_m_va_cp.html#c_m_va_cp">Cross product</a><span class="headline-id">c.m.va.cp</span>

  1414. 

  1415. </li>

  1416. <li>

  1417. <a href="./c_m_va_tp.html#c_m_va_tp">Triple Products</a><span class="headline-id">c.m.va.tp</span>

  1418. 

  1419. </li>

  1420. <li>

  1421. <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>

  1422. 

  1423. </li>

  1424. 

  1425. </ul>

  1426. </details>

  1427. </li>

  1428. <li>

  1429. 

  1430. <details>

  1431. <summary>

  1432. <a href="./c_m_dc.html#c_m_dc">Differential Calculus</a><span class="headline-id">c.m.dc</span>

  1433. 

  1434. 

  1435. </summary>

  1436. <ul>

  1437. <li>

  1438. <a href="./c_m_dc_g.html#c_m_dc_g">Gradient</a><span class="headline-id">c.m.dc.g</span>

  1439. 

  1440. </li>

  1441. <li>

  1442. <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>

  1443. 

  1444. </li>

  1445. <li>

  1446. <a href="./c_m_dc_div.html#c_m_dc_div">The Divergence</a><span class="headline-id">c.m.dc.div</span>

  1447. 

  1448. </li>

  1449. <li>

  1450. <a href="./c_m_dc_curl.html#c_m_dc_curl">The Curl</a><span class="headline-id">c.m.dc.curl</span>

  1451. 

  1452. </li>

  1453. <li>

  1454. <a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>

  1455. 

  1456. </li>

  1457. <li>

  1458. <a href="./c_m_dc_d2.html#c_m_dc_d2">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span>

  1459. 

  1460. </li>

  1461. 

  1462. </ul>

  1463. </details>

  1464. </li>

  1465. <li>

  1466. 

  1467. <details>

  1468. <summary>

  1469. <a href="./c_m_ic.html#c_m_ic">Integral Calculus</a><span class="headline-id">c.m.ic</span>

  1470. 

  1471. 

  1472. </summary>

  1473. <ul>

  1474. <li>

  1475. <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>

  1476. 

  1477. </li>

  1478. <li>

  1479. <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>

  1480. 

  1481. </li>

  1482. <li>

  1483. <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>

  1484. 

  1485. </li>

  1486. <li>

  1487. <a href="./c_m_ic_gauss.html#c_m_ic_gauss">Gauss' Theorem</a><span class="headline-id">c.m.ic.gauss</span>

  1488. 

  1489. </li>

  1490. <li>

  1491. <a href="./c_m_ic_stokes.html#c_m_ic_stokes">Stokes' Theorem</a><span class="headline-id">c.m.ic.stokes</span>

  1492. 

  1493. </li>

  1494. <li>

  1495. <a href="./c_m_ic_ip.html#c_m_ic_ip">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span>

  1496. 

  1497. </li>

  1498. 

  1499. </ul>

  1500. </details>

  1501. </li>

  1502. <li>

  1503. 

  1504. <details>

  1505. <summary>

  1506. <a href="./c_m_cs.html#c_m_cs">Coordinate Systems</a><span class="headline-id">c.m.cs</span>

  1507. 

  1508. 

  1509. </summary>

  1510. <ul>

  1511. <li>

  1512. <a href="./c_m_cs_sph.html#c_m_cs_sph">Spherical Coordinates</a><span class="headline-id">c.m.cs.sph</span>

  1513. 

  1514. </li>

  1515. <li>

  1516. <a href="./c_m_cs_cyl.html#c_m_cs_cyl">Cylindrical Coordinates</a><span class="headline-id">c.m.cs.cyl</span>

  1517. 

  1518. </li>

  1519. <li>

  1520. <a href="./c_m_cs_hyp.html#c_m_cs_hyp">Hyperbolic Coordinates</a><span class="headline-id">c.m.cs.hyp</span>

  1521. 

  1522. </li>

  1523. 

  1524. </ul>

  1525. </details>

  1526. </li>

  1527. <li>

  1528. 

  1529. <details>

  1530. <summary>

  1531. <a href="./c_m_dd.html#c_m_dd">Dirac delta Distribution</a><span class="headline-id">c.m.dd</span>

  1532. 

  1533. 

  1534. </summary>

  1535. <ul>

  1536. <li>

  1537. <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>

  1538. 

  1539. </li>

  1540. <li>

  1541. <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>

  1542. 

  1543. </li>

  1544. <li>

  1545. <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>

  1546. 

  1547. </li>

  1548. 

  1549. </ul>

  1550. </details>

  1551. </li>

  1552. <li>

  1553. 

  1554. <details>

  1555. <summary>

  1556. <a href="./c_m_vf.html#c_m_vf">Vector Fields</a><span class="headline-id">c.m.vf</span>

  1557. 

  1558. 

  1559. </summary>

  1560. <ul>

  1561. <li>

  1562. <a href="./c_m_vf_helm.html#c_m_vf_helm">The Helmholtz Theorem</a><span class="headline-id">c.m.vf.helm</span>

  1563. 

  1564. </li>

  1565. <li>

  1566. <a href="./c_m_vf_pot.html#c_m_vf_pot">Potentials</a><span class="headline-id">c.m.vf.pot</span>

  1567. 

  1568. </li>

  1569. 

  1570. </ul>

  1571. </details>

  1572. </li>

  1573. <li>

  1574. 

  1575. <details>

  1576. <summary>

  1577. <a href="./c_m_uf.html#c_m_uf">Useful Formulas</a><span class="headline-id">c.m.uf</span>

  1578. 

  1579. 

  1580. </summary>

  1581. <ul>

  1582. <li>

  1583. <a href="./c_m_uf_cyl.html#c_m_uf_cyl">Cylindrical coordinates</a><span class="headline-id">c.m.uf.cyl</span>

  1584. 

  1585. </li>

  1586. <li>

  1587. <a href="./c_m_uf_sph.html#c_m_uf_sph">Spherical coordinates</a><span class="headline-id">c.m.uf.sph</span>

  1588. 

  1589. </li>

  1590. <li>

  1591. <a href="./c_m_uf_vi.html#c_m_uf_vi">Vector identities</a><span class="headline-id">c.m.uf.vi</span>

  1592. 

  1593. </li>

  1594. 

  1595. </ul>

  1596. </details>

  1597. </li>

  1598. 

  1599. </ul>

  1600. </details>

  1601. </li>

  1602. 

  1603. </ul>

  1604. </details>

  1605. </li>

  1606. 

  1607. </ul>

  1608. </details>

  1609. </nav>

  1610. <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:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul><div id="outline-container-ems_ca_fe_L" class="outline-5">

  1611. <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">

  1612.   <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"/>

  1613.   <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"/>

  1614. </svg></a><span class="headline-id">ems.ca.fe.L</span></h5>

  1615. <div class="outline-text-5" id="text-ems_ca_fe_L">

  1616. <p>

  1617. In regions of space where there is no charge density,

  1618. the potential must solve Laplace's equation.

  1619. Let us discuss how solutions to this equation look,

  1620. in increasingly complicated situations.

  1621. </p>

  1622. </div>

  1623. 

  1624. 

  1625. <div id="outline-container-ems_ca_fe_L_1d" class="outline-6">

  1626. <h6 id="ems_ca_fe_L_1d"><a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a></h6>

  1627. <div class="outline-text-6" id="text-ems_ca_fe_L_1d">

  1628. <p>

  1629. In one dimension, the potential is a single-variable

  1630. function \(\phi (x)\) and the Laplace equation reads

  1631. </p>

  1632. 

  1633. <div class="eqlabel" id="orgd38ad90">

  1634. <p>

  1635. <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">

  1636.   <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"/>

  1637.   <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"/>

  1638. </svg></a>

  1639. </p>

  1640. <div class="alteqlabels" id="org23686dc">

  1641. 

  1642. </div>

  1643. 

  1644. </div>

  1645. <p>

  1646. \[

  1647. \frac{d^2 \phi(x)}{dx^2} = 0.

  1648. \tag{Lap_1d}\label{Lap_1d}

  1649. \]

  1650. </p>

  1651. 

  1652. <p>

  1653. The solution to this is

  1654. </p>

  1655. <div class="eqlabel" id="org78665cb">

  1656. <p>

  1657. <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">

  1658.   <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"/>

  1659.   <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"/>

  1660. </svg></a>

  1661. </p>

  1662. <div class="alteqlabels" id="orge0dbdb5">

  1663. <ul class="org-ul">

  1664. <li>Gr (3.6)</li>

  1665. </ul>

  1666. 

  1667. </div>

  1668. 

  1669. </div>

  1670. <p>

  1671. \[

  1672. \phi(x) = a x + b

  1673. \tag{Lap_1d_sol}\label{Lap_1d_sol}

  1674. \]

  1675. </p>

  1676. 

  1677. <p>

  1678. Properties:

  1679. </p>

  1680. <ul class="org-ul">

  1681. <li>

  1682. <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

  1683. the region where Laplace is satisfied, of course).</li>

  1684. <li>

  1685. <b>No extrema</b>: \(\phi(x)\) has no local extrema. Max/min

  1686. values must occur at boundaries.</li>

  1687. </ul>

  1688. 

  1689. 

  1690. <p>

  1691. In a particular problem, to fix the solution (said

  1692. 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

  1693. conditions. Concretely, for a finite segment,

  1694. a solution exists and is unique if one is

  1695. provided with any of these possibilities:

  1696. </p>

  1697. 

  1698. <ul class="org-ul">

  1699. <li>\(\phi\) at both boundaries</li>

  1700. <li>\(\phi\) and \(\frac{d\phi}{dx}\) at one boundary</li>

  1701. <li>\(\phi\) at one boundary, \(\frac{d\phi}{dx}\) at the other.</li>

  1702. </ul>

  1703. 

  1704. <p>

  1705. Specifying \(\frac{d\phi}{dx}\) at both boundaries

  1706. provides insufficient information, since you get

  1707. an inconsistency if the derivatives don't match.

  1708. </p>

  1709. </div>

  1710. </div>

  1711. 

  1712. 

  1713. <div id="outline-container-ems_ca_fe_L_2d" class="outline-6">

  1714. <h6 id="ems_ca_fe_L_2d"><a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a></h6>

  1715. <div class="outline-text-6" id="text-ems_ca_fe_L_2d">

  1716. <p>

  1717. In two dimensions, the potential becomes a function

  1718. of two variables (here: \(x\) and \(y\)), so Laplace's

  1719. equation now reads

  1720. </p>

  1721. <div class="eqlabel" id="orgd4e7f28">

  1722. <p>

  1723. <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">

  1724.   <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"/>

  1725.   <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"/>

  1726. </svg></a>

  1727. </p>

  1728. <div class="alteqlabels" id="orga64f9ad">

  1729. 

  1730. </div>

  1731. 

  1732. </div>

  1733. 

  1734. \begin{equation*}

  1735. \frac{\partial^2 \phi (x,y)}{\partial x^2}

  1736. + \frac{\partial^2 \phi (x,y)}{\partial y^2} = 0.

  1737. \tag{Lap_2d}\label{Lap_2d}

  1738. \end{equation*}

  1739. 

  1740. <p>

  1741. Properties:

  1742. </p>

  1743. <ul class="org-ul">

  1744. <li>

  1745. <b>Balance</b>: \(\phi(x,y)\) equals the average value around the point:</li>

  1746. </ul>

  1747. <p>

  1748. \[

  1749. \phi(x,y) = \frac{1}{2\pi R} \oint dl ~\phi

  1750. \]

  1751. </p>

  1752. <ul class="org-ul">

  1753. <li>

  1754. <b>No extrema</b>: \(\phi\) has no local maxima or minima.  All extrema occur at the boundaries.</li>

  1755. </ul>

  1756. </div>

  1757. </div>

  1758. 

  1759. 

  1760. <div id="outline-container-ems_ca_fe_L_3d" class="outline-6">

  1761. <h6 id="ems_ca_fe_L_3d"><a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a></h6>

  1762. <div class="outline-text-6" id="text-ems_ca_fe_L_3d">

  1763. <p>

  1764. In three dimensions, we will write the potential

  1765. as a function of a 3-dimensional vector, \(\phi({\bf r})\).

  1766. The Laplace equation is (we repeat)

  1767. </p>

  1768. 

  1769. <p>

  1770. \[

  1771. {\boldsymbol \nabla}^2 \phi ({\bf r}) = 0

  1772. \]

  1773. </p>

  1774. 

  1775. <p>

  1776. <b>Theorem</b>: if \(\phi\) satisfies Laplace, then its value at

  1777. a point equals its value averaged over a sphere

  1778. \(S_R({\bf r})\) of any radius \(R\) centered on this point

  1779. (and of course not containing any charges),

  1780. </p>

  1781. <div class="eqlabel" id="org598b2e6">

  1782. <p>

  1783. <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">

  1784.   <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"/>

  1785.   <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"/>

  1786. </svg></a>

  1787. </p>

  1788. <div class="alteqlabels" id="orgdb48832">

  1789. 

  1790. </div>

  1791. 

  1792. </div>

  1793. <p>

  1794. \[

  1795. \phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_R({\bf r})} da' ~\phi ({\bf r}')

  1796. \tag{p_ball_avg}\label{p_ball_avg}

  1797. \]

  1798. </p>

  1799. 

  1800. <details id="orgd997cbe">

  1801. <summary id="org6b22e3f">

  1802. <strong>Physicist's proof</strong>

  1803. </summary>

  1804. <p>

  1805. Consider a sphere of radius \(R\) centered at the origin

  1806. carrying charge \(q\) spread with a uniform surface charge density over its surface. Bring in a point charge \(q'\) from

  1807. infinity up to a distance \(R'\) (with \(R' &gt; R\)) from the center

  1808. of the sphere.

  1809. </p>

  1810. 

  1811. <p>

  1812. We know that the field created by the sphere coincides

  1813. with that of a point charge \(q\) at the origin.

  1814. The work required to bring the \(q'\) charge into position is thus

  1815. simply \(W = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.

  1816. </p>

  1817. 

  1818. <p>

  1819. We can however proceed the other way: fixing \(q'\) in place,

  1820. and then bringing the charged sphere into position;

  1821. the work (energy) has to coincide with our previous result.

  1822. But this energy is now given by the integral of the

  1823. potential \(\phi_{q', {\bf r'}}\)

  1824. created by \(q'\) (sitting at \({\bf r'}\)) over the sphere

  1825. times the surface charge density on the sphere,

  1826. namely

  1827. </p>

  1828. 

  1829. <p>

  1830. \[

  1831. W = \oint_{S_R} da ~\sigma ~\phi_{q', {\bf r}'} ({\bf r})

  1832. \]

  1833. </p>

  1834. 

  1835. <p>

  1836. But \(\sigma = q/4\pi R^2\) and is a constant over the

  1837. sphere, so \(W = q \times \frac{1}{4\pi R^2} \oint da ~\phi_{q', {\bf r}'} ({\bf r})\).

  1838. </p>

  1839. 

  1840. <p>

  1841. Equating this with the previous results shows that

  1842. </p>

  1843. 

  1844. <p>

  1845. \[

  1846. \frac{q'}{4\pi \varepsilon_0 R'} = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})

  1847. \]

  1848. but this also equals the potential at \({\bf r} = 0\) created by the charge

  1849. \(q'\) at \({\bf r'}\), <i>i.e.</i> \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\).

  1850. In other words, we have thus shown that for the potential created by a single point

  1851. charge \(q'\) at \(R'\), the value at a point (here the origin)

  1852. coincides with the value averaged over a sphere

  1853. or an arbitrary radius \(R\) centered on the same point.

  1854. </p>

  1855. 

  1856. <p>

  1857. By the principle of superposition, this works for an

  1858. arbitrary distribution of charges outside the sphere,

  1859. proving the theorem.

  1860. </p>

  1861. </details>

  1862. 

  1863. <details id="org2ff5eec">

  1864. <summary id="orgc0446a1">

  1865. <strong>Formal proof</strong>

  1866. </summary>

  1867. 

  1868. <p>

  1869. Consider a function \(f({\bf r})\) and its average over

  1870. a ball of radius \(R\) centered on \({\bf r}\):

  1871. </p>

  1872. 

  1873. <p>

  1874. \[

  1875. f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R ({\bf r})} da' ~ f ({\bf r}')

  1876. \]

  1877. </p>

  1878. 

  1879. <p>

  1880. In spherical coordinates defined around the point \({\bf r}\),

  1881. we have \(da' = R^2 sin \theta d\theta d\varphi \equiv R^2 d\Omega\).

  1882. Differentiating with respect to \(R\),

  1883. </p>

  1884. 

  1885. <p>

  1886. \[

  1887. \frac{d}{dR} f_{S_R} = \frac{1}{4\pi} \oint_{S_R} d\Omega ~\left.\frac{\partial f}{\partial r}\right|_{r=R}

  1888. \]

  1889. </p>

  1890. 

  1891. <p>

  1892. with \(f\) differentiated with respect to the radial coordiate.

  1893. We can rewrite this by noting that \(R^2 d\Omega ~\hat{\bf r}\)

  1894. is the normal differential surface area \(d{\bf a}\), while

  1895. \(\left.\frac{\partial f}{\partial r}\right|_{r=R}\) is the radial component of the gradient

  1896. of \(f\) in spherical coordinates. Thus,

  1897. </p>

  1898. 

  1899. <p>

  1900. \[

  1901. \frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \oint_{S_R} d{\bf a} \cdot ~\nabla f

  1902. \]

  1903. </p>

  1904. 

  1905. <p>

  1906. Invoking the divergence theorem and using the definition

  1907. of the Laplacian operator \(\nabla^2 = \nabla \cdot \nabla\),

  1908. we get the following general

  1909. </p>

  1910. 

  1911. <p>

  1912. <b>Theorem</b>:

  1913. </p>

  1914. <div class="eqlabel" id="orge37257a">

  1915. <p>

  1916. <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">

  1917.   <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"/>

  1918.   <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"/>

  1919. </svg></a>

  1920. </p>

  1921. <div class="alteqlabels" id="orgb3a7f07">

  1922. 

  1923. </div>

  1924. 

  1925. </div>

  1926. 

  1927. <p>

  1928. \[

  1929. \frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \int_{V_R} d\tau ~\nabla^2 f

  1930. \tag{dfdR_intLap}\label{dfdR_intLap}

  1931. \]

  1932. </p>

  1933. 

  1934. 

  1935. <p>

  1936. For the electrostatic potential away from charges, we have

  1937. \[

  1938. \nabla^2 \phi = 0 ~\rightarrow \frac{d}{dR} \phi_{S_R} = 0

  1939. \]

  1940. namely the ball average is independent of the ball size.

  1941. Since the value at the center is simply the average for

  1942. an infinitesimally small ball, we get the result announced above.

  1943. </p>

  1944. </details>

  1945. 

  1946. <p>

  1947. <b>Theorem (Earnshaw, mathematical version)</b>: \(\phi\) has no local extrema except at the boundaries.

  1948. </p>

  1949. 

  1950. <p>

  1951. <b>Proof</b>: write the second derivatives as

  1952. </p>

  1953. 

  1954. <p>

  1955. \[

  1956. \frac{\partial^2 \phi({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}

  1957. \frac{\partial^2 \phi({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}

  1958. \frac{\partial^2 \phi({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}

  1959. f_x + f_y + f_z = 0.

  1960. \]

  1961. </p>

  1962. 

  1963. <p>

  1964. The \(f_a ({\bf r})\) represent the three components of the curvature of \(\phi({\bf r})\).

  1965. An extremum of \(\phi\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} \phi |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)

  1966. for any infinitesimal displacement \(\delta{\bf r}\) around the extremum point.  For a local

  1967. minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 \phi}{\partial r_i \partial r_j} \delta r_i \delta r_j &gt; 0\)

  1968. for any displacement vector.  Choosing alternately displacements along the three axes,

  1969. the form becomes \(f_x (\delta x)^2\), \(f_y (\delta y)^2\) or \(f_z (\delta z)^2\).  Since the squared displacements

  1970. are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f_z &gt; 0\).  This is impossible in view

  1971. of the \(f_x + f_y + f_z = 0\) condition above.

  1972. </p>

  1973. 

  1974. <div class="eqlabel" id="org39eb572">

  1975. <p>

  1976. <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">

  1977.   <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"/>

  1978.   <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"/>

  1979. </svg></a>

  1980. </p>

  1981. <div class="alteqlabels" id="org2cc4d11">

  1982. 

  1983. </div>

  1984. 

  1985. </div>

  1986. <div class="info div" id="orgcd59178">

  1987. <p>

  1988. <b>Earnshaw's theorem (physical version)</b> <br>

  1989. </p>

  1990. 

  1991. <p>

  1992. It is impossible to find a static distribution of charges which generates an electrostatic field

  1993. displaying a stable equilibrium position in empty space.

  1994. </p>

  1995. 

  1996. </div>

  1997. 

  1998. <p>

  1999. Going back to Poisson's equation, we can make a few comments:

  2000. </p>

  2001. 

  2002. <ul class="org-ul">

  2003. <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>

  2004. 

  2005. <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>

  2006. 

  2007. <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>

  2008. </ul>

  2009. 

  2010. 

  2011. <p>

  2012. We therefore want to ask the question:  <i>under what conditions can an electrostatic problem be fully

  2013. defined by solving Poisson's equation?</i>

  2014. </p>

  2015. 

  2016. <p>

  2017. We start by mentioning some cases, and interpreting them thereafter.

  2018. </p>

  2019. 

  2020. <p>

  2021. <b>Charge density is known throughout space</b>: in this case,

  2022. the electrostatic potential is uniquely determined

  2023. by Poisson's equation, which

  2024. is explicitly solved by <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>.

  2025. One can explicitly verify this:

  2026. </p>

  2027. 

  2028. <p>

  2029. \[

  2030. {\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}'|}

  2031. = \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}.

  2032. \]

  2033. </p>

  2034. 

  2035. <p>

  2036. 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

  2037. 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>

  2038. converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved.

  2039. </p>

  2040. 

  2041. <p>

  2042. <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

  2043. in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density

  2044. \(\rho ({\bf r})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),

  2045. and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\).

  2046. Of course, \({\cal S}\) need not be a connected surface.

  2047. </p>

  2048. 

  2049. 

  2050. <p>

  2051. <b>"Known boundary potential" case: charge density in closed volume and potential at boundary are both known</b>:  the electrostatic potential is uniquely determined

  2052. in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density

  2053. \(\rho ({\bf r})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the

  2054. boundary \({\cal S}\).  Of course, \({\cal S}\) need not be a connected surface.

  2055. </p>

  2056. 

  2057. <p>

  2058. Here, the logic is quite simple:  since the electrostatic potential is known on all the surface enclosing

  2059. the space \({\cal V}\), and since Poisson's equation is local, we need not consider anything outside of \({\cal V}\)

  2060. to obtain \(V\) within \({\cal V}\).

  2061. </p>

  2062. 

  2063. <p>

  2064. Given a solution \(V_1 ({\bf r})\), we can easily show that it is unique.  Suppose there was another solution

  2065. \(V_2 ({\bf r})\).  Look at the difference, \(U \equiv V_1 - V_2\).  In the bulk, \(U\) obeys the Laplace

  2066. equation

  2067. </p>

  2068. 

  2069. <p>

  2070. \[

  2071. {\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0.

  2072. \]

  2073. </p>

  2074. 

  2075. <p>

  2076. Moreover, \(U ({\bf r}) = 0\) for \({\bf r} \in {\cal S}\).  Since solutions to the Laplace equation take

  2077. their maximal and minimal value on the boundary, we must have \(U = 0\) \(\forall {\bf r} \in {\cal V}\)

  2078. </p>

  2079. 

  2080. 

  2081. <p>

  2082. 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?

  2083. </p>

  2084. </div>

  2085. </div>

  2086. </div>

  2087. 

  2088. 

  2089. 

  2090. <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul>

  2091. <br>

  2092. <hr>

  2093. <div class="license">

  2094. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  2095. target="_blank" class="m-2">

  2096. <img alt="Creative Commons License" style="border-width:0"

  2097. src="https://licensebuttons.net/l/by/4.0/80x15.png"/>

  2098. </a>

  2099. Except where otherwise noted, all content is licensed under a

  2100. <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"

  2101. target="_blank">Creative Commons Attribution 4.0 International License</a>.

  2102. </div>

  2103. <div id="postamble" class="status">

  2104. <p class="author">Author: Jean-Sébastien Caux</p>

  2105. <p class="date">Created: 2022-03-24 Thu 08:42</p>

  2106. <p class="validation"></p>

  2107. </div>

  2108. 

  2109. 

  2110. </div>

  2111. </html>